. 

ARITHMETIC 


M  AN 


^ 


LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA. 


GIFT  OF 


Class 


Pacific  Theological  Seminary, 


PRACTICAL 


BUSINESS  ARITHMETIC, 


COMMON  SCHOOLS  AND  ACADEMIES. 


INCLUDING   A   GBEAT  VARIETY   OF 


PKOMISCUOUS    EXAMPLES. 


BY 

WHITMAN    PECK,    A.M., 

ItFTHOR   OF   THE  PROMISCUOUS  EXERCISES  IK  ANDREW'S  LATIN   LESSONS 
(REVISED   EDITION.) 


NEW    YOEK: 

J.  W.  SCHEEMEEHOEN  &  CO.,  PUBLISHEES, 

14  BOND    STEEET. 

1868. 


V? 


Entered,  according  to  Act  of  Congress,  in  the  year  1868,  by 
W.    PECK, 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States,  for  the 
Southern  District  of  New  York. 


H.  C.  STOOTHOFF, 

STEAM  BOOK  AND  JOB  PBINTER, 

39  and  41  Centre  St.,  N.  T. 


PREFACE. 


THE  distinguishing  feature  of  this  Arithmetic,  that  which  has 
chiefly  led  to  its  publication,  is  its  containing,  in  addition  to  the  ex- 
amples under  each  rule,  a  large  number  of  "  Promiscuous  Examples  " 
under  several  different  rules.  No  two  of  these  together  being  alike, 
pupils  need  to  think  how  each  one  is  to  be  done  independently  of 
another,  instead  of  only  doing  all  like  one  already  done  in  the  book, 
or  by  their  teacher.  They  can  often  do  page  after  page  of  examples 
as  commonly  arranged  under  their  respective  rules,  though  they 
eould  not  do  much  simpler  examples  as  they  are  apt  to  occur  in  prac- 
tical business.  Hence  men  often  say,  that  their  knowledge  of  Arith- 
metic, when  they  commenced  business,  consisted  in  little  more  than 
knowing  how  to  add,  subtract,  multiply  and  divide,  when  directed  to 
do  so  in  an  arithmetic,  or  by  their  teacher.  This  defect,  it  is  be- 
lieved, will  be  remedied  by  the  repeated  use  of  the  promiscuous  ex- 
amples in  this  book.  They  are  so  classified  and  arranged  that  each 
"Exercise"  requires  the  application  of  what  has  been  previously 
studied  in  some  portion  of  the  book.  The  author  having  found 
such  exercises  almost  indispensable  in  teaching  arithmetic,  has 
thought  it  would  be  a  great  convenience  to  teachers,  to  have  an 
arithmetic  containing  a  large  number  of  promiscuous  examples. 
Many  experienced  teachers,  also,  all  with  whom  he  has  consulted, 
have  confirmed  him  in  this  opinion.  Hence,  though  there  is  per- 
haps too  great  a  variety  of  arithmetics  already  in  use,  this  is  offered 
to  the  public. 

It  is  believed,  too,  that  this  arithmetic  contains  in  one  book,  all  the 
most  important  matter  usually  found  in  an  arithmetical  series,  in 
which  much  the  same  matter  is  repeated  in  different  books,  thus 
greatly  increasing  the  expense,  without  any  real  advantage.  The 
first  part,  including  the  Fundamental  Rules,  is  adapted  to  children 
beginning  to  study  arithmetic  after  having  received  a  little  oral  in- 
struction; and  they  are  advanced  so  gradually,  that  they  will  be  apt 
to  learn  this  part  thoroughly  before  they  reach  Compound  Numbers 
and  Fractious 

111900 


PREFACE. 

Most  of  the  examples  in  this  arithmetic  are  designedly  short,  that 
less  time  may  be  consumed  in  the  operations,  and  more  be  devoted 
to  learning  the  principles  and  their  applications.  They  are,  also,  so 
simple  that  most  pupils  may  be  expected  to  do  them,  witla  a  little  as- 
sistance in  some  cases,  without  requiring  too  much  of  the  teacher's 
time  in  explaining  what  they  seldom  understand  or  remember.  Some 
more  difficult  examples,  designed  for  advanced  pupils,  will  be  found 
at  the  end  of  the  book,  and  it  is  designed  to  publish  in  another  book 
many  more  such  examples,  and  some  principles  of  Higher  Arith- 
metic omitted  in  this,  which,  however,  is  sufficient  to  fit  persons  for 
the  practical  business  of  life. 

The  author,  also,  thinks  that  he  has  greatly  simplified  the  study 
of  arithmetic  by  reducing  the  number  of  its  rules.  He  applies  the 
Bules  for  Reduction  of  Compound  Numbers  to  Reduction  of  Frac- 
tional Compound  Numbers  (common  and  decimal,)  and  the  rules  of 
Percentage  to  all  its  various  applications,  such  as  Commission, 
Brokerage,  Stocks,  Profit  and  Loss,  etc.,  etc. 

Suggestions  to  Teachers. 

Pupils  should  be  required  to  explain  fully  the  examples  in  arith- 
metic, at  least  enough  of  them  to  show  that  they  thoroughly  under- 
stand them.  At  first,  they  will  need  to  use  the  blackboard  or  then- 
slates,  but  they  should  also  learn  to  give  the  explanations  mentally, 
omitting  the  numbers  if  too  large  to  be  thus  calculated,  but  naming 
them  at  each  step  as  they  proceed.  If  they  can  do  this  beforehand, 
they  need  not  be  required  to  perform  operations  with  which  they  are 
already  perfectly  familiar.  In  this  way  they  will  study  mental  as 
well  as  written  arithmetic. 

Though  the  Promiscuous  Examples  are  numerous,  some  pupils 
may  need  to  do  them  repeatedly,  in  order  to  become  as  familiar  as 
they  ought  to  be  with  the  practical  application  of  what  they  have 
previously  studied.  Others  may  not  need  to  do  them  all.  One  or 
two  exercises  at  a  time  may  be  sufficient.  After  a  few  days,  give 
them  one  or  two  more  similar  exercises,  and  continue  to  do  this 
from  time  to  time  till  the  principles  and  rules  are  permanently  fixed 
in  their  minds. 

The  rules  are  designed  to  aid  pupils  in  making  their  own  rules, 
rather  than  to  be  verbally  committed  to  memory.  They  should  learn 
to  perform  all  arithmetical  operations,  and  explain  them,  inde- 
pendently of  the  rules  in  books. 


CONTENTS. 


NUMBER 

NOTATION  (Roman) . . 
Arabic 

NUMERATION 

FUNDAMENTAL  RULES 


PAGE 

,      7 

,      7 

8 

10 
14 


ADDITION 15 

SUBTRACTION 22 

MULTIPLICATION 27 

By  Composite 

Numbers 35 

DIVISION 38 

Short 39 

Long 45 

By  Composite  Numbers  47 

General  Principles 49 

PROMISCUOUS  EXAMPLES  in  Ad- 
dition, Subtraction,  Multi- 
plication and  Division 50 

UNITED  STATES  MONET 56 

Table— Aliquot  Parts 57 

Promiscuous  Examples. .  68 

Bills ' 74 

COMPOUND  NUMBERS 77 

MONEY — English  or  Sterling  77 

WEIGHTS— Troy,  Table 77 

Avoirdupois,  Table  78 

Apothecaries,  Table  78 

Miscellane's,  Table  78 

MEASURES— Cloth,  Table 78 

Long,  Table 79 

Surveyor's,  Table  79 


PAGE 

MEASURES— Square,  Table ...  79 
Cubic,  Table  ...  80 
Wine,  Table....  81 
Beer,  Table  ....  82 

Dry,  Table 82 

Time,  Table  ...     82 
Circular,  Table  .     84 
Miscellaneous     Table     of 

Units,  &c.,  Paper,  Books    84 
REDUCTION     of      Compound 

Numbers 85 

Examples 88—95 

Promiscuous  Examples  95—105 
Addition    of      Compound 

Numbers 105 

Subtraction  of  Compound 

Numbers 106 

Multiplication     of     Com- 
pound Numbers 108 

Division     of     Compound 

Numbers 109 

Longitude  and  Time 110 

Promiscuous  Examples. . .  112 

Cancellation 115 

Prime  and  Composite  Num- 
bers    116 

Greatest  Common  Divisor  118 
Least  Common  Multiple . .  119 

FRACTIONS 121 

Common 122 

Reduction  of  . .  .  125 


CONTENTS. 


Addition  of 131 

Subtraction  of 132 

Multiplication  of 134 

Division  of 136 

Pronrscuous  Examples 

139—146 

DECIMAL  FEACTIONS 146 

Addition  of 148 

Subtraction  of 150 

Multiplication  of 151 

Division  of 153 

Promiscuous  Examples  . . .  156 
Beduction     of     Common 

Fractions  to  Decimals  . .  159 
Beduction  of  Decimal  Frac- 
tions to  Common 160 

Fractional  Compound  Num- 
bers    161 

Promiscuous  Examples . . .  164 
Promiscuous  Examples  in 
Common    and    Decimal 

Fractions 166 

DUODECIMALS 177 

ANALYSIS 180 

PEECENTAGE 184 

Commission 190 

Brokerage 191 

Stocks 191 

Gold 192 

Insurance 192 

Profit  and  Loss 1 

Interest 198 

Partial  Payments 203 


PAGE 

Compound  Interest 209 

Discount 211 

Bank 213 

Taxes 215 

Duties 217 

Exchange 218 

Partnership 223 

Promiscuous  Examples  in 
the  various  applications 

of  Percentage 226—236 

EQUATION  OF  PAYMENTS 236 

KEDUCTION  OF  CUBBENCIES  . .  241 

EATIO 244 

PROPOETION 

Compound 2 

Conjoined 250 

ALLIGATION 251 

INVOLUTION 2 

EVOLUTION 

Square  Koot 255 

Cube  Boot 

PEOGBESSION — Arithmetical..  264 
Geometrical . .  256 

MENSUEATION 267 

PEOMISCUOUS    EXAMPLES  — 
U.    S.  Money    and    Com- 
pound Numbers 271 

Fractions,    Common    and 

Decimal 274 

Percentage  and  its  applica- 
tions   279 

Miscellaneous  Eules 284 


ARITHMETIC 


Article  1.— Arithmetic  is  the  science  of  numbers.  It 
teaches  their  nature  and  use. 

Number  is  one  or  more  things,  or  Units  ;  as  one,  two, 
three  ;  the  number  of  pupils  in  a  class  is  four,  five,  &c. 

Abstract  numbers  are  numbers  not  applied  to  any  par- 
ticular thing ;  as  one,  two,  five,  &c.  Concrete  numbers 
are  numbers  applied  to  particular  things  ;  as  five  men, 
ten  cents. 


NOTATION. 

Art  2. — Notation  is  the  method  of  writing  numbers. 

There  are  two  methods,  the  Roman,  introduced  by  the 
ancient  Romans,  and  the  Arabic,  introduced  by  the 
Arabians,  which  is  chiefly  used  in  Arithmetic. 

Art.  3* — The  Roman  method  uses  letters  for  numbers  ; 
as,  I,  one;  V,  five;  X,  ten;  L,  fifty;  C,  one  hundred;  D, 
five  hundred;  M,  one  thousand. 

These  seven  letters  repeated  or  united  express  all  other 
numbers. 

If  a  letter  is  repeated,  its  value  is  multiplied  as  many 
times  ;  as,  II,  (two  times  one,)  two;  XX,  twenty;  XXX, 
thirty. 


8  NOTATION. 

If  a  letter  is  written  before  another  of  greater  value, 
its  value  is  subtracted  from  that  of  the  greater  ;  but  if 
written  after  another  of  greater  value  it  is  added;  as,  IV, 
four;  VI,  six;  IX,  nine;  XI,  eleven. 

A  small  line  ( — )  over  a  letter  multiplies  its  value  a 
thousand  times  ;  as,  V,  five  thousand. 

TABLE   OF  ROMAN  LETTERS  USED  FOB  NUMBERS. 

I.  One.  IX.  Nine.         LXXX.  Eighty. 

IE.  Two.  X.  Ten.  XG.  Ninety. 

m.  Three.  XX.  Twenty.  C.  One  hundred. 

IV.  Four.  XXX.  Thirty.  CO.  Two  hundred. 

V.  Five.  XL.  Forty.  D.  Five  hundred. 

VI.  Six.  L.  Fifty.  M.  One  thousand 

VII.  Seven.  LX.  Sixty.  V.  Five  thousand. 

VIH.  Eight.  LXX.  Seventy. 

Art.  4. — The  Arabic  Notation  uses  the  following  ten 
figures  for  numbers  : 

(Written]      O.     /.    2.    3.     6.    5.    6.      /.    (9.  #. 

Naught  or  o;Qe  ^Q  three,  four.  five.  six.  seven,  eight,  nine. 
Cipher. 

(Printed)     0.     1.     2.      3.      4       5.     6.     7.      8.      9. 

These  figures,  except  the  cipher,  are  called  Digits* 
A  figure  written  alone,  or   on  the  right  hand  of   a 
number,  has  only  its  simple  value;  as,  1,  one;  2,  two;  5, 
five,  &c. 

A  figure  written  before  another  has  ten  times  its  simple 
value;  also  when  prefixed  to  two  others  it  has  one  hun- 
dred times  its  simple  value.  Hence  figures  increase  in 
value  ten  fold  from  right  to  left. 


NOTATION.  9 

10  (ten  and  naught)  ten.  20  (two  tens)  twenty. 

11  (ten  and  one)  eleven.  21  (2  tens  and  1)  twenty-one. 

12  (ten  and  two)  twelve.  30  (three  tens)  thirty.  &c. 

13  (ten  and  three)  thirteen.  40  (four  tens)  forty,  &c. 

14  (ten  and  four)  fourteen.  50  (five  tens)  fifty,  &c. 

15  (ten  and  five)  fifteen.  60  (six  tens)  sixty,  &c. 

16  (ten  and  six)  sixteen.  70  (seven  tens)  seventy,  &c. 

17  (ten  and  seven)  seventeen.  80  (eight  tens)  eighty,  &c. 

18  (ten  and  eight)  eighteen.  90  (nine  tens)  ninety,  &c. 

19  (ten  and  nine)  nineteen.  100  ( ten  tens)  one  hundred,  &c. 

In  all  the  numbers  from  10 — 19  the  figure  1  is  used  for  ten.  In  the 
number  11  the  figure  1  is  used  for  ten  and  one  ;  and  in  111  it  is  used 
for  one  hundred,  ten  and  one. 

Next  to  hundreds  are  thousands,  tens  of  thousands, 
hundreds  of  thousands,  millions,  &c.,  as  in  the  following 
French  method,  which  is  chiefly  used. 

FRENCH  NOTATION  AND  NUMERATION  TABLE. 


Next  to  trillions  are  quadrillions,  quintillions,  sextillions,  sept.il- 
lions,  octillions,  nonillions,  decillions,  &c. 

In  this  table  numbers  are  divided  into  periods  of  three 
figures  each,  beginning  at  the  right  hand,  the  1st  units, 
the  2d  thousands,  the  3d  millions,  &c. 


10  NUMEBATION. 


ENGLISH   NOTATION   AND   NUMERATION  TABLE. 


11 

CM      0 
O      O 

rj        QQ 

5    H    H    fi    H 
20987          65432 


Periods. 


RULE  FOR  NOTATION. — Leaving  space  enough  on  the  right 
for  as  many  periods,  of  three  figures  each,  as  the  number 
will  contain,  begin  at  the  left  hand,  and  write  the  number 
belonging  to'  each  period,  filling  the  vacant  places  with 
ciphers. 

EXAMPLE. — Write  two  millions,  seventy-five  thousand, 
three  hundred  and  five. 

There  will  be  two  periods  on  the  right  of  millions.  Write  2  in  the 
millions'  period,  075  in  the  thousands'  period,  and  305  in  the  last  or 
units'  period  ;  thus,  2,075,305. 


NUMERATION. 

Art,  5.— Numeration  is  reading  numbers. 

Small  numbers  are  easily  read  by  repeating  the  name 
of  each  figure  as  it  is  written.  In  reading  a  large  num- 
ber observe  the  following 

RULE. — Consider  the  number  as  divided  into  periods  of 
three  figures  each,  beginning  at  the  right  hand  ;  then,  begin- 


NUMERATION. 


11 


ning  at  the  left  hand,  read  each  period  as  if  it  stood  alone, 
adding  its  name,  except  that  of  the  last ;  thus, 

The  number  1,230,987,654,321,  is  read  one  trillion,  two  hundred 
and  thirty  billions,  nine  hundred  and  eighty-seven  millions,  six  hun- 
dred and  fifty-four  thousand,  three  hundred  and  twenty-one. 


EXEKCISES  IN   NUMERATION. 


Bead  the  following  numbers  down  and  across  the  page.  It 
will  be  best  for  pupils  to  write  them  first,  if  they  have  not 
learned  to  do  so  readily  and  plainly. 


10 

28 

30 

48 

13 

25 

33 

45 

16 

22 

36 

42 

19 

27 

39 

47 

11 

24 

31 

44 

14 

21 

34 

41 

17 

29 

37 

49 

12 

26 

32 

46 

15 

23 

35 

43 

50 

61 

72 

80 

91 

56 

68 

76 

88 

98 

59 

65 

73 

85 

95 

51 

62 

79 

82 

92 

54 

67 

70 

87 

97 

57 

64 

74 

84 

94 

52 

60 

77 

81 

99 

55 

69 

75 

89 

96 

58 

66 

78 

86 

93 

18   20   38 


40 


53   63   71   83 


90 


210 

228 
234 
389 
465 


328 
333 

456 
598 
55C 


430 

550 

-  672 

761 

891 

445 

678 

789 

890 

901 

543 

785 

876 

983 

779 

671 

872 

963 

753 

861 

655 

741 

833 

922 

766 

990 
985 

742 
888 


1000 
1234 

2345 
3456 
4567 
5678 


10000 
23456 
34567 
45678 
56789 
67890 


100000 
345678 
456789 
567890 
678901 
789012 


1000000 
4567890 
5678901 
6789012 
7890123 
8901234 


10000000 

123456789 

2345678901 

34567890123 


1000000000 

12345678901 

345678901234 

4567890123456 


12,345,678,908,765,432,102,468. 


12  EXERCISES   IN  NOTATION. 


IN  NOTATION. 

Write  all  the  numbers  from — 
Ten  to  twenty-five.  Fifty  to  seventy-five. 

Twenty-five  to  fifty.  Seventy-five  to  one  hundred. 

Write— 

One  hundred  and  ten.  Five  hundred  and  sixty-seven. 

Two  hundred  and  eleven.  Six  hundred  and  seventy-eight. 

Three  hundred  and  one.  Seven  hundred  and  eighty-nine. 

Four  hundred  and  twenty.        Eight  hundred  and  eight. 
Five  hundred  and  sixty-seven.  Nine  hundred  and  ninety. 
Six  hundred  and  seventy-nine.  Ten  hundred  and  twenty. 
Eight  hundred  and  ninety.        Twelve  hundred  and  eleven. 
Nine  hundred  and  thirty-four.  Sixteen  hundred  and  seventeen. 
Ten  hundred  and  eleven.  Eighteen  hundred  and  ninety. 

Eleven  hundred  and  twenty.      Nine  hundred  and  seventy-five. 
Twelve  hundred  and  fifty-five.  Eight  hundred  and  sixty-four. 
Fifteen  hundred  and  sixty-two.  Seven  hundred  and  fifty-three. 
Nine  hundred  and  eighty -six.    Six  hundred  and  forty-two. 
Six  hundred  and  fifty-four.        Five  hundred  and  thirty  one. 
Three  hundred  and  twenty-one.  Four  hundred  and  twenty. 
One  hundred  and  twenty-three.  Three  hundred  and  one. 
Four  hundred  and  fifty-six.       Two  hundred  and  three. 
Seven  hundred  and  eight.          Three  hundred  and  fourteen. 
Nine  hundred  and  ten.  Four  hundred  and  twenty-four. 

Two  hundred  and  eleven.          Five  hundred  and  fifteen. 
Three  hundred  and  forty-five.  Six  hundred  and  ten. 
Four  hundred  and  fifty-six.       Seven  hundred  and  twelve. 

One  hundred. 

Two  thousand. 

Thirty  thousand. 

Four  hundred  thousand. 

Five  millions. 

Six  hundred  and  six. 

Seven  thousand  eight  hundred  and  nine. 


NOTATION.  18 

Eighty  thousand  and  ninety. 
Nine  hundred  thousand  and  one  hundred. 
Ten  million,  eleven  thousand  and  twelve. 
Thirteen  hundred  and  fourteen. 
Fifteen  thousand,  one  hundred  and  two. 
Three  hundred  thousand  and  four. 
Sixty  million,  seventy  thousand  and  eight  hundred. 
One  hundred  and  ten  millions,  two  hundred  and  thirty-four 
thousand,  four  hundred  and  five. 

Two  hundred  and  thirty-four. 

Five  thousand,  six  hundred  and  seventy-eight. 

Ninety  thousand  and  seventeen. 

Three  hundred  thousand,  five  hundred  and  seven. 

Eleven  millions,  one  hundred  and  five  thousand. 

Five  hundred  millions,  seven  thousand  and  eighty-one. 

Seventy-five  thousand,  three  hundred  and  forty. 

Eight  hundred  thousand,  two  hundred  and  five. 

Nine  thousand,  seven  hundred  and  fifty-three. 

Three  millions,  four  hundred  and  thirty-two. 

Twelve  millions,  eleven  thousand  and  nine  hundred. 

One  hundred  and  twenty  millions,  seventeen  thousand,  six 
hundred  and  seven. 

Six  thousand,  seven  hundred  and  thirty-one. 

Seven  hundred  and  forty-eight. 

Sixty-eight  thousand,  four  hundred  and  fifty-one. 

Thirty-nine  millions,  nine  hundred  and  twelve  thousand, 
three  hundred  and  ninety-six. 

Seven  hundred  and  fifty  thousand,  five  hundred  and  sixty- 
three. 

Forty-six  thousand,  five  hundred  and  four. 

Twelve  hundred  and  ninety-seven. 

Two  thousand,  five  hundred  and  sixty-six. 

Four  millions,  five  hundred  and  four  thousand,  three  hun- 
dred and  twenty-two. 

Twenty-five  thousand,  seven  hundred  and  thirty-eight. 

One  thousand,  four  hundred  and  thirty-three. 


FUNDAMENTAL  RULES. 

Five  millions,  three  hundred  and  one  thousand,  seven  hun- 
dred and  ninety-five. 

The  following  are  not  designed  for  very  young  pupils. 
Write— 

One  billion,  two  hundred  and  thirty-four  millions,  five  thou- 
sand and  seven  hundred. 

Three  trillions,  twenty-five  billions,  three  hundred  and  four 
millions,  forty-five  thousand,  six  hundred  and  seventy-four. 

Twenty  billions,  four  hundred  and  twelve  millions,  sixty- 
five  thousand  and  thirty-two. 

Four  hundred  trillions,  seventy-  seven  billions,  seven  hun- 
dred and  seven  millions,  nine  thousand,  five  hundred  and 
sixty-three. 

Four  quadrillions  and  five  hundred  trillions. 

Five  quintillions  and  sixty-eight  trillions. 

Six  sextillions  and  five  hundred  quintillions. 

Seventy  billions. 

Eighty  trillions. 

Ninety  quadrillions. 

One  hundred  quintillions,  two  hundred  and  ten  quadrillions, 
thirty-five  trillions,  seven  hundred  billions  and  sixty-four 
millions. 

Fifteen  sextillions,  five  hundred  and  sixty  quintillions,  four 
hundred  and  twenty-five  trillions. 


FUNDAMENTAL    RULES. 

Art.  6. — Arithmetic  teaches  the  use  of  numbers  in  four 
principal  ways,  viz  :  Addition,  Subtraction,  Multiplica- 
tion, and  Division,  called  the  Fundamental  Rules  of 

Arithmetic. 


ADDITION.  15 


ADDITION. 

j^rt.  7. — Addition  is  uniting  two  or  more  numbers  in 
one.  The  number  thus  found,  or  the  answer,  is  called 
the  Sum  or  Amount. 

Simple  Addition  is  uniting  like  numbers,  or  numbers  of 
the  same  name,  in  one  ;  as,  3  apples  added  to  4  apples 
are  7  apples. 

Unlike  numbers  cannot  be  added  ;  as  3  apples  and  4  pears  are 
neither  7  apples  nor  7  pears. 

Addition  is  often  expressed  by  the  sign  (-J-)  Plus, 
placed  between  numbers  to  be  added. 

[The  sign  (=)  of  equality  placed  between  numbers,  shows  that 
they  are  equal.  ] 

ILLUSTRATION. — The  sum  or  amount  of  2  added  to  3  is  equal  to  5  ; 
2+3=5. 

ADDITION  TABLE. 

[This  table  is  promiscuously  arranged.  The  answers  are  not  given, 
because  it  is  better  that  pupils  should  learn  them  by  thinking  for 
themselves,  and  not  have  them  for  reference.  They  should  be  able 
to  recite  them  perfectly  and  promptly.] 


H!  i  i.S  ! 


2120323 
2203130 
Sums  1   2 

133042414344 
323414340424 

052545153550 
515350525456 

626461636566 
163656267606 

173757727476 
727476773577 


16  ADDITION  OF  UNITS. 

Add 


J  2 

8 

4 

8 

6 

8 

8 

8 

8 

5 

8 

7 

1    8 

3 

8 

5 

8 

7 

8 

8 

4 

8 

6 

8 

3 

9 

5 

9 

7 

9 

9 

4 

9 

6 

9 

8 

9 

4 

9 

6 

9 

8 

9 

9 

5 

9 

7 

9 

ADDITION    OF    UNITS. 

MENTAL  EXERCISES. 

How  many  boys  are  2  boys  and  1  more  ?  1+2  ?  2+2  ?«, 
3+1  ?  2+3  ?  3+4  ?  4+2  ?  3+3  ?  4+3  ?  4+4  ? 

How  many  girls  are  5  girls  and  2  more  ?  2+5  ?  3+5  ? 
5+4?  5+3?  5+5?  6+3?  4+6?  6+5?  6+6? 

How  many  men  are  7  men  and  3  more  ?  3+7  ?  7+4  ? 
5+7  ?  7+6  ?  7+7  ?  8+3  ?  4+9  ?  5+8  ?  9+5  ?  6+8  ? 
8+8?  7+9?  9+9? 

How  many  women  are  8  women  and  2  more  ?  3+8  ?  9+4  ? 
8+5?  6+9?  8+7?  9+6? 

EXAMPLES  FOE  THE   SLATE   OE  BLACKBOARD. 

4 

EXAMPLE  1.— Add  4,  3,  2,  3,  5,  and  2. 

1  Process. — 2  and  5  are  7,  and  3  are  10,  and  2  are 

12,  and  3  are  15,  and  4  are  19 .    Name  only  the  result  K 

of  each  addition  ;  as  7,  10,  12,  15,  19. 

^  Ans.  19 

EULE. —  Write  the  numbers  under  one  another;  draw  a 
line  underneath^  and  under  it  write  the  sum  or  amount. 

EXAMPLES. 

(2)      (3)      (4)      (5)      (6)      (7)      (8)       (9)     (10)    (11)  (12)  (13) 


2 

3 

4 

5 

4 

3 

2 

4 

5 

4 

2 

3 

3 

4 

5 

1 

3 

3 

2 

4 

5 

5 

3 

4 

4 

3 

3 

5 

1 

3 

2 

4 

5 

3 

4 

3 

3 

4 

4 

2 

2 

3 

2 

4 

5 

4 

4 

2 

4 

2 

2 

5 

3 

3 

2 

4 

5 

5 

3 

3 

2 

1 

4 

3 

4 

3 

2 

4 

5 

3 

2 

4 

3 

1 

1 

5 

4 

2 

3 

5 

5 

4 

3 

5 

ADDITION.  17 

<14)  (15)  (16)  (17)       (18)  (19)    (20)    (21)     (22)  (23)  (24)     (25) 


4 

5 

1 

2 

3 

1 

2 

3 

4 

5 

6 

7 

3 

1 

5 

4 

4 

2 

3 

2 

1 

1 

1 

2 

4 

5 

1 

2 

5 

3 

4 

3 

4 

5 

6 

7 

2 

2 

4 

3 

2 

4 

5 

4 

2 

2 

2 

1 

4 

5 

1 

2 

2 

5 

6 

3 

4 

5 

6 

7 

1 

3 

3 

4 

1 

6 

7 

5 

3 

3 

3 

3 

4 

5 

5 

4 

5 

7 

8 

3 

4 

5 

6 

7 

6 

7 

7 

4 

4 

4 

5 

(26) 

(27) 

(28) 

(29) 

(30) 

(31) 

(32) 

(33) 

(34) 

(35) 

(36) 

(37) 

1 

2 

3 

4 

5 

6 

7 

1 

2 

3 

7 

1 

7 

6 

4 

4 

6 

6 

6 

2 

3 

7 

7 

2 

2 

3 

5 

5 

7 

6 

7 

3 

4 

5 

7 

3 

7 

6 

6 

5 

5 

6 

6 

4 

4 

3 

7 

4 

3 

4 

7 

6 

6 

6 

7 

5 

6 

7 

7 

5 

7 

6 

6 

6 

7 

6 

6 

6 

7 

3 

7 

6 

4 

5 

4 

5 

4 

6 

7 

7 

0 

6 

7 

7 

7 

7 

2 

5 

3 

6 

6 

4 

6 

7 

7 

8 

(38) 

(39) 

(40) 

(41) 

(42)  (- 

43)  (44 

)  (4£ 

>)  (46; 

1  (47) 

(48) 

(49) 

(50) 

2 

3 

4 

5 

6 

7       8 

9 

7 

8 

9 

6 

5 

3 

4 

5 

4 

7 

8       9 

0 

8 

6 

4 

3 

8 

4 

5 

6 

6 

8 

9       0 

8 

9 

4 

8 

9 

3 

5 

6 

7 

8 

9 

0       8 

1 

7 

7 

3 

6 

7 

6 

7 

8 

9 

0 

7       9 

9 

0 

3 

5 

3 

9 

7 

8 

9 

0 

2 

8       1 

8 

6 

2 

7 

9 

8 

8 

0 

0 

8 

3 

9       8 

2 

4 

6 

8 

6 

0 

9 

9 

8 

7 

4 

7       9 

8 

9 

8 

7 

3 

8 

ADDITION  OF  UNITS,  TENS,  HUNDREDS,  &o. 

MENTAJJ   EXEECISES. 

How  many  lambs  are  10  lambs  and  1  more  ?  10+3  ? 

10+5  ?  10+7  ?  10+9  ?  10+2  ?  10+4  ?  10+6  ?  10+8  ? 

11+1?  11+2?  11+4?  11+6?  11+8?  11+3?  11+5? 
11+7  ?  11+9  ? 

How  many  sheep  are  12  sheep  and  1  more  ?  12+3  ?  12+5  ? 

12+7?  12+9?  12+2?  12+4?  12+6?  12+8-?  13+3? 
13+4?  13+6?  13+8?  13+5?  13+7?  13+9? 


18  KULE  OF  ADDITION. 

How  many  horses  are  14  horses  and  2  more  ?  14+4  ? 

14+6?  14+8?  14+3?  14+5?  14+7?  14+9?  15+2? 

15+4?  15+6?  15+8?  15+3?  15+5?  15+7?  15+9? 
16+2  ?  16+5  ?  16+8  ? 

How  many  cows  are  17  cows  and  3  more  ?  17+5  ?  17+7  ? 

17+9?  17+4?  17+6?  17+8?  18+4?  18+6?  18+8? 

18+9?  18+7?  13+5?  19+3?  19+5?  19+7?  19+9? 
19+6  ?  19+8  ? 

EXAMPLES   FOR   THE   SLATE. 

EXAMPLE  51.— Add  123,  234,  345,  and  456. 

Process. — Write  the  numbers  thus, 


Ans.  1158 

Add  the  right  hand  column  6+5+4+3=18  units,  or  1  ten  and  8 
units.  Write  8  under  units  and  add  1  to  the  tens.  Add  the  second 
column  1+5+4+3+2=15  tens,  1  hundred  and  5  tens.  Write  5 
under  tens  and  add  1  to  hundreds.  Thus  proceed. 

RULE. —  Write  the  numbers  under  one  another,  so  that  all 
the  right-hand  figures  shall  be  in  the  same  column,  and  the 
others  in  proper  order,  tens  next  to  units,  &e. 

Beginning  at  the  right  hand,  add  each  column  separately. 
If  the  sum  consists  of  only  one  figure,  write  it  under  the 
column ;  but  if  it  consists  of  two  or  more,  write  only  the 
right  hand  figure  and  carry  or  add  the  others  to  the  next 
column  if  there  is  any;  otherwise  write  both  figures. 

PROOF. — Add  the  same  columns  downward. 

Figures  of  different  local  value  cannot  be  added  ;  2  tens  and  3  units 
are  neither  5  tens  (50)  r*or  5,  but  23. 

EXAMPLES. 

(52)  (53)  (54)  (55)  (56)  (57)  (58)  (59)  (60)  (61)  (62)  (63)  (64) 

23  34  45  50  01  12  23  32  45  33  44  55  34 

34  51  01  12  23  45  44  34  54  44  44  55  53 

45  02  23  34  34  30  54  22  45  55  44  55  45 

23  34  45  51  50  44  32  10  54  22  44  55  34 

34  51  04  23  12  32  45  54  45  11  44  55  53 


ADDITION.  19 

(65)  (66)  (67)  (68)  (69)  (70)  (71)  (72)  (73)  (74)  (75)  (76)  .(77)  (78) 

12  23  34  45  56  67  78  89  90  66  77  88  99  89 

34  45  56  67  78  89  90  01  12  66  77  88  99  98 

56  67  78  89  90  01  12   23  34  66  77  88  99  79 

78  89  90  01  12  23  34  45  56  66  77  88  99  96 

90091823  34^556677866  77889945 

(79)  (80)   (81)  (82)   (83)   (84)   (85)  (86)   (87)  (88) 

123   423   345  456   567   678   789  890   901  910 

456   456   678  789   890   901   012  123   234  315 

789   789   901  '012   123   234   345  456   567  678 

012   045   234  345   456   567   678  789   890  910 

345   678   567  678   789   890   901  012   123  234 

678   997   891  901   012   123   234  345   456  567 

(89)   (90)   (91)   (92)   (93)    (94)   (95)   (96)  (97) 


123 

234 

345 

431 

543 

654 

764 

876 

987 

456 

567 

678 

098 

210 

321 

325 

543 

654 

789 

890 

901 

765 

987 

098 

109 

210 

321 

123 

123 

234 

321 

654 

765 

876 

987 

098 

456 

564 

567 

098 

321 

432 

543 

654 

765 

789 

897 

890 

765 

098 

109 

210 

321 

431 

123 

135 

123 

432 

765 

876 

987 

098 

098 

456 

678 

456 

109 

431 

543 

632 

765 

726 

(98) 

(99) 

(100) 

(101) 

(102) 

(103) 

(104) 

(105) 

(106) 

789 

890 

901 

678 

567 

456 

345 

231 

123 

012 

123 

234 

901 

890 

789 

678 

456 

345 

345 

456 

567 

234 

123 

012 

901 

678 

567 

678 

789 

£90 

567 

456 

334 

223 

890 

789 

901 

012 

123 

890 

789 

455 

344 

012 

901 

234 

345 

456 

123 

Oil 

667 

556 

234 

123 

567 

678 

789 

456 

223 

788 

677 

456 

345 

890 

901 

012 

789 

334 

990 

889 

678 

507 

123 

234 

345 

012 

455 

Oil 

901 

890 

789 

456 

567 

678 

345 

667 

223 

334 

123 

901 

EXAMPLES  FOE  THE  SLATE. 


MENTAL    EXERCISES. 

How  many  chickens  are  20  chickens  and  2  more  ?  20  -{-  4  ? 
20+7  ?  20+9  ?  21+3  ?  21+5  ?  21+7  ?  21+9  ?  22+4  ?  22+6  ? 
22+8  ?  23+3  ?  23+4  ?  23+7  ?  23+9  ?  23+6  ? 

How  many  ducks  are  24  ducks  and  3  more  ?  24+6  ?  24+9  ? 
25+4  ?  25+7  ?  25+9  ?  26+8  ?  27+4  ?  28+7  ?  29+5  ?  30+7  ? 
31+8  ?  32+6  ?  33+5  ?  36+9  ?  39+8  ?  41+5  ?  43+7  ?  46+8  ? 
47+7?  49+8?  51+7?  55+5?  57+6?  58+7? 

How  many  pigeons  are  63  pigeons  and  9  more?  63+7  ?  64+5  ? 
65+7  ?  66+6  ?  67+7  ?  68+8  ?  69+9  ?  70+7  ?  75+6  ?  76+8  ? 
77+8?  79+4?  79+6?  79+7?  81+9?  83+£  ?  84+9?  86+7? 
88+8  ?  90+9  ?  91+7  ?  93+8  ?  97+5  ?  98+7  ?  99+9  ? 

How  many  quails  are  10  quails  and  10  more  ?  10  + 12  ? 
10+17?  10+19?  11+13?  11+15?  12+11?  13+12?  14+13? 
15+12?  16+11?  17+13?  18+15?  19+11?  19+14?  19+16? 
19+19  ? 


EXAMPLES  FOB  THE  SLATE. 

(107) 

(108) 

(109) 

(110) 

(111) 

(112) 

12345 

23456 

34567 

45678 

56789 

67899 

67890 

78901 

89012 

90987 

98765 

12344 

11223 

23456 

34567 

65432 

56789 

45677 

34455 

78901 

89012 

10123 

91023 

78900 

€6778 

23456 

34567 

45678 

34567 

01234 

89900 

78901 

89012 

90876 

78901 

56789 

11223 

23456 

34567 

54321 

12345 

98765 

34455 

78901 

89012 

09876 

67890 

54321 

(113) 

(114) 

(115) 

(116) 

(117) 

(118) 

34682 

46820 

65431 

76432 

61234 

5064 

57931 

06842 

76543 

67675 

45678 

785 

24680 

3697 

87654 

56567 

54321 

9543 

13564 

568 

98764 

78789 

76428 

6748 

2805 

7634 

09876 

97973 

85947 

97054 

3579 

86420 

1234 

46467 

64758 

7865 

46820 

13579 

765 

33590 

81927 

753 

97531 

24680 

12076 

45678 

79635 

39 

ADDITION.  21 

119.  65340  +  6731  +  748  +  68451+396+7503+46075  + 1290+ 
25738+46803. 

120.  54268+405+1708  +  43671+72049  +  492+1760+25357+ 
1434+84162. 

121.  246768  +  21380  +  4075  +  126849  +  257  + 1305  +  24350+ 
439871+40306+601734. 

122.  3947+73845+300901+499091+45131+564429+484292+ 
178737+58072+65344+194532+758+14. 

Find  the  sum  of  the  following  numbers  : 

EXAMPLE  (123.) 

One  hundred  and  eighteen  thousand,  nine  hundred  and 
forty-eight. 

One  thousand,  one  hundred  and  ninety-two. 

Two  millions,  eight  hundred  and  sixteen  thousand,  seven 
hundred  and  sixty. 

Ninety  thousand,  four  hundred  and  forty -five. 

One  hundred  and  twenty-eight  thousand. 

One  million,  one  hundred  and  forty-three  thousand,  eight 
hundred  and  twelve. 

Twenty-four  thousand,  six  hundred  and  sixty-four. 

Three  millions,  two  hundred  and  forty-three  thousand. 

EXAMPLE    (124.) 

Twenty-seven  millions,  six  hundred  and  nineteen  thousand^ 
eight  hundred  and  sixty-six. 

Three  hundred  and  fifty-four  thousand,  seven  hundred  and 
ninety-seven. 

Two  minions,  two  hundred  and  ninety  thousand,  three  hun- 
dred and  sixty-three. 

Nine  hundred  and  thirty  thousand,  four  hundred  and  thirty. 

Four  hundred  thousand. 

One  hundred  and  twenty-seven  millions,  seven  hundred  and 
seventy-eight  thousand,  nine  hundred  and  eighty-one. 

One  million,  four  hundred  and  twenty-one  thousand,  six 
hundred  and  sixty-one. 

Eight  hundred  and  sixty-nine  thousand.  ^ 


22  EXAMPLES. 

EXAMPLE    (125.) 

Sixty  millions,  seven  hundred  and  eight  thousand,  five  hun- 
dred and  two. 

Two  millions,  nine  hundred  and  thirty-seven  thousand  and 
sixty-six. 

Sixty-one  thousand. 

One  million,  six  hundred  and  twenty-five  thousand. 

EXAMPLE  (126.) 

Two  millions,  one  hundred  and  twenty  thousand,  three  hun- 
dred and  ninety-seven. 
Eighty-six  thousand. 
Two  millions  and  six  hundred  thousand. 
Four  hundred  and  fifty  thousand. 

PBACTICAIj  EXAMPLES. 

127.  A  farmer  has  sold  at  different  times  30,  45,  48,  50,  56, 
and  63  bushels  of  oats  ;  how  many  bushels  altogether  ? 

128.  A  family  consumed  in  a  year  6  loads  of  coal,  weighing 
as  foUows  :  1250,  1168,  987,  1076,  879,  and  1275  pounds;  how 
many  pounds  in  all  ? 

129.  A  merchant  has  bought  5  cases  of  muslin,  containing 
respectively  the  following  number  of  yards  :  963,  897,  985, 
1005,  and  889  ;  how  many  yards  in  all  ? 

130.  The  population  of  the  New  England  States  in  1850  was 
respectively  583,000,  318,000,  314,000,  995,000,  148,000,  and 
371,000,  what  was  the  whole  population  of  New  England  ? 


SUBTRACTION. 

Art,  8. — Subtraction  is  taking  a  less  number  from  a 
greater.  The  number  thus  found  is  called  the  Differ- 
ence or  Remainder. 

The  number  to  be  subtracted  is  called  the  Subtra- 


SUBTRACTION.  23 

hend,  and  the  number  from  whicH  it  is  to  be  taken  the 
Minuend. 

Simple  Subtraction  is  taking  one  number  from  another 
of  the  same  name. 

Subtraction  is  sometimes  expressed  by  the  sign  ( — ) 
minus*  The  number  written  after  the  sign  is  to  be  sub- 
tracted from  the  one  before  it ;  as  5 — 3=2  ;  three  sub- 
tracted from  5  leaves  2. 

SUBTRACTION  TABLE. 
[It  is  promiscuously  arranged  and  without  answers.] 

From 123412342342 

Take 0      1      2      0      1      1      0      2      1      1      2      1 

Remainder     1      1   &c. 

454564565676 
122313411234 

787878787878 
132537425466 

9    10      9     10      9    10      9     10      9     10      9    10 
234567823456 

11    12    11     12    11     12     11     12    11     12    11    12 
694725839876 

13    14    13    14    13     14    13     14    13     14    13    14 
987654328769 

15    16    15    16    15    16    15    16    15    16    15    16 
896798754346 


24  MENTAL  EXERCISES. 

Prom 17    18     17*   18    17    18    17    18    17    18    17    17 

Take 9      6      3      8      5      7      8      9      6      5      4      3 

19    18    17    16    19    18    17    16    15    14    18     19 
978986989897 

MENTAL  EXEKCISES. 

In  a  class  of  2  girls  how  many  will  be  left  if  1  girl  leave  it  ? 
2  girls  ?  How  many  are  2—1  ?  2—2  ? 

In  a  class  of  3  boys  how  many  will  be  left  if  2  boys  leave  it? 
1  boy  ?  3  boys  ? 

How  many  are  3—2  ?  3—1  ?  3—3  ? 

Edward  has  5  apples  and  his  sister  3,  how  many  more  has  he 
than  his  sister  ? 

How  many  are  5 — 4  ?  5 — 2  ?  5 — 5  ?  5 — 3  ?  5—1  ? 

Mary  had  6  cakes  and  gave  3  to  her  brother,  how  many  had 
she  left  ? 

How  many  are  6—5 ?  6—3?  6—1?  6—6?  6—4?  6—2? 

Anna  has  7  good  marks  at  school  and  4  bad  ones,  how  many 
more  good  ones  has  she  than  bad  ones  ? 

How  many  are  7—3  ?  7—5  ?  7—1  ?  7—7  ?  7—6  ?  7—4  ? 

John  had  8  oranges  and  has  given  away  all  except  5,  how 
many  has  he  given  away  ? 

How  many  are  8—5  ?  8—7?  8—3  ?  8—6  ?  8—2  ?  8-4  ? 

Thomas  has  9  cents  and  his  brother  7,  how  many  more  has 
he  than  his  brother  ? 

How  many  are  9—6  ?  9—3  ?  9—7  ?  9—4  ?  9—5  ? 

EXAMPLES  FOB  THE  STATE  OB  BLACKBOABD. 

EXAMPLE  1.— From  5698  subtract  3245. 


Process. — 5  units  from  8  units  leave   3  units  ;  4  tens 
from  9  tens  leave  5  tens,  &c. 

Ans.  2453 
Ex.  2— From  7653  subtract  4865. 

Process.— Since  5  cannot  be  taken  from  3,  add  one  of 
the  5  tens,  or  10  to  the  3  ;  then  5  from  13  leave  8.     For 
the  same  reason  add  one  of  the  6  hundreds,  or  10  to  the      Ans.  2988 
4  tens  left,  and  take  6  from  14,  &c. 


SUBTRACTION.  ZO 

Ex.  3. —From  6004  subtract  3125. 

Process. — Though  there  are  no  tens  in  the  upper  num- 
ber, it  contains  at  least  10  units  ;  therefore  add  ten  to 
the  4  and  take  5  from  14,  and  consider  that  only  9  of  the      Ans.  2579 
10  tens  which  the  number,  contains  remain  in  the  tens 
place  ;  or  the  result  is  the  same  if  1  is  added  or  earned  to  the   next 
lower  figure,  the  two  numbers  being  equally  increased. 

RULE. —  Write  the  less  number  under  the  greater,  so  that 
the  right  hand  figure  of  each  shall  be  under  each  other,  and 
draw  a  line  beneath  them. 

Begin  at  the  right  hand  and  subtract  each  figure  of  the 
lower  number  from  the  one  directly  above  it,  and  write  the 
difference  beneath  it. 

If  the  upper  figure  is  less  than  the  lower,  add  ten  to  it,  then 
subtract  and  carry  one  to  the  next  lower,  or  subtract  one  from 
the  next  upper  figure. 

PROOF. — Add  the  remainder  to  the  less  number,  and  if  the 
sum  is  equal  to  the  greater,  the  subtraction  is  correct. 

EXAMPLES. 

W        (5)         (6)        (7)        (8) 
654321      765432      876543      987654     987653 
331201      234201      145312      431212     534443 

(9)        (10)        (11)        (12)        (13) 
767676     878787     989898     888888     999999 
423122      534231      675643      544332     497531 


(14) 
7532175321 
4826084356 

(15) 
8642066420 
3736393035 

(16) 
9753195713 
2846098648 

(17) 
1023678415 
897534706 

(18) 
2125374859 
958763465 

(19) 
7685946354 
5869370819 

(20) 
8796059384 
3948126893 

(21) 
9281736471 
2736453673 

2 

26 


EXAMPLES. 


(22) 

(23)          (24) 

(25) 

3435796354 

4356534798     5768743276 

65439S7672 

1527697616 

2785374689     2378679863 

5734689778 

(26) 

(27)          (28) 

(29) 

7569354978 

8954978654     9872579846 

7963864201 

3678567889 

5379889910     6743287938 

3576383712 

(30) 

(31)          (32) 

(33) 

6743876549 

9576854378     4759813548 

8743645211 

4371968732 

2768907569     1964679473 

1170927802 

(34) 

(35)          (36) 

(37) 

5478698472 
3789589756 

6802468024     7791357915 
3579137951     4749454347 

8642042864 
7894013498 

(38) 

(39)          (40) 

(41) 

6547328964 
2785679546 

5473143256     8321731214 
1758916407     5732157321 

9543682410 
5437691607 

(42) 

(43)          (44) 

(45) 

4002007100 
870345076 

5002070022     6070005000 
3754327819     3700072001 

8573000012 
4781973607 

46. 

75694  —  3590  =  what  number  ? 

47. 

87532  —   7615  •= 

48. 

987563—  43215=     " 

49. 

107923—  36001=     " 

50. 

100000  —   999  =     " 

51. 

845067—  71389=     " 

52. 

1789543—  76508=     " 

53. 

200100—  54321=     " 

54. 

500000—  99887=     " 

55. 

6000000  —  887766  =     " 

56.  From  ten  thousand  and  five,  subtract  seven  thousand 
five  hundred  and  seven. 

57.  From  twenty-five  thousand,  one  hundred  and  six,  sub- 
tract ten  thousand,  six  hundred  and  seventy-nine. 


MULTIPLICATION.  27 

58.  From  one  hundred  thousand  and  fifty-eight,  subtract 
fifteen  thousand,  three  hundred  and  ninety. 

59.  From  one  million,  subtract  seven  thousand  and  five. 

60.  From  ten  millions,  ten  thousand  and  ten,  subtract  five 
hundred  thousand,  five  hundred  and  five. 

PBACTICAL   EXAMPLES. 

61.  A  man  purchased  a  farm  for  6750  dollars,  and  sold  it  for 
9000  dollars,  how  much  did  he  gain  ? 

62.  Mt.  Washington,  N.  H.,  is  6285  feet  high  ;  Mt.  Mans- 
field, Vt.,  4,279  feet  ;  how  much  higher  is  Mt.  Washington 
than  Mt.  Mansfield  ? 

63.  The  Missouri  River  is  3100  miles  long,  and  the  Missis- 
sippi 2500  ;  how  much  longer  is  the  former  than  the  latter  ? 

64.  The  population  of  London  in  1850  was  two  millions^ 
three  hundred  and  sixty-three  thousand  ;  that  of  Paris  one 
million,  fifty-three  thousand  and  two  hundred  ;  what  was  the 
difference  ? 

65.  A  man  was  born  in  the  year  1780  and  died  in  1859,  how 
old  was  he  when  he  died  ? 

66.  North  America  contains  eight  millions  of  square  miles  ; 
Europe,  three  millions  five  hundred  thousand;  how  many  more 
square  miles  are  there  in  North  America  than  in  Europe  ? 


MULTIPLICATION. 

Art.  9. — Multiplication  is  finding  a  number,  equal  to  a 
given  number  repeated  by  addition  as  many  times  as 
there  are  units  in  another  given  number ;  as,  4  times  6 
are  24,  or  6+6+6+6=24. 

The  number  to  be  multiplied  is  called  the  Multiplicand* 
The  number  by  which  the  other  is  multiplied  is  called 
the  Multiplier*  The  answer,  or  number  found  by  multi- 
plication, is  called  the  Product* 


28 


MULTIPLICATION. 


Both  the  multiplicand  and  multiplier  are  called  Factors* 
because  they  make  the  product. 

Simple  multiplication  is  that  in  which  the  multiplicand 
is  of  only  one  name. 

The  sign  of  multiplication  is  (X)  an  oblique  cross. 

ILLUSTRATION.— 5  times  6  are  30  :  6X5=30.  The  multiplicand  is  6  ; 
the  multiplier  5  ;  and  the  product  30.  5  and  6  are  also  factors. 

The  multiplicand  must  properly  be  of  the  same  name  as  the  pro- 
duct or  answer  required,  though  it  is  often  more  convenient  to  make 
the  larger  number  the  multiplicand,  while  the  result  is  the  same. 
The  number  of  marks  below  is  the  same,  whether  they  are  arranged 
in  five  groups  of  six  each,  or  six  groups  of  five  each,  or  all  in  one 
group  ;  thus 


Both  6X5  and  5X6=30.  But  if  6  men  can  reap  a  field  in  5  days, 
and  it  be  required  to  find  how  many  men  could  reap  it  in  one  day, 
the  multiplicand  will  be  6  men,  and  the  product  30  men  ;  while  if  it 
be  required  to  find  in  how  many  days  one  man  could  reap  it,  the 
multiplicand  will  be  5  days,  and  the  product  30  days. 

The  multiplier,  though  often  used  as  a  concrete  number  applied  to 
some  particular  things,  is  properly  only  an  abstract  number.  In  the 
illustration  of  the  preceding  remark,  6  men  multiplied  by  5  days,  or 
6  days  multiplied  by  6  men,  would  be  absurd  ;  but  6  men  multiplied 
by  5,  which  is  the  same  as  the  number  of  days,  are  30  men  ;  and  5 
days  multiplied  by  6,  which  is  the  same  as  the  number  of  men,  are 
30  days. 

MUI/TIPUCATION  TABLE. 


2  times 

3  times 

4  times 

5  times 

6  times 

7  times 

1    are   2 

1   are   3 

1   are   4 

1    are   5 

1   are   6 

1  are     7 

2    "      4 

2 

6 

2    "     8 

2 

10 

2 

12 

2    "    14 

3    "      6 

3 

9 

3 

12 

3 

15 

3 

18 

3    "    21 

4    "      8 

4 

12 

4 

16 

4 

20 

4 

24 

4 

28 

5    "    10 

5 

15 

5 

20 

5 

25 

5 

30 

5 

36 

6    "    12 

6 

18 

6 

24 

6 

30 

6 

36 

6 

42 

7    "    14 

7 

21 

7 

28 

7 

35 

7 

42 

7 

49 

8    "    16 

8 

24 

8 

32 

8 

40 

8 

48 

8 

56 

9    "    18 

9 

27 

9 

36 

9 

45 

9 

54 

9 

63 

10    "    20 

10 

30 

10 

40 

10 

50 

10 

60 

10 

70 

11    "    22 

11 

33 

11 

44 

11    "    55 

11 

66 

11    "     77 

12    "    24 

12 

36 

12    "    48 

12    "    60 

12    "    72 

12    "    84 

MULTIPLICATION. 


29 


8  times 

9  times 

10  times 

11  times 

12  times 

1  are  8 

1  are  9 

1  are  10 

1  are  11 

1  are  12 

2 

16 

2 

18 

2 

20 

•2 

22 

2 

24 

3 

24 

3 

27 

3 

30 

3 

33 

3 

36 

4 

32 

4 

36 

4 

10 

4 

44 

4 

48 

5 

40 

5 

45 

5 

50 

5 

5 

60 

6 

48 

6 

54 

6 

GO 

6 

66 

6 

72 

7 

56 

7 

63 

7 

70 

7 

77 

7 

8 

64 

8 

72 

8 

80 

8 

88 

8 

96 

9 

72 

9 

81 

9 

90 

9 

99 

9 

10S 

10 

80 

10 

90 

10 

100 

10 

110 

10 

1-20 

11 

88 

11 

99 

11 

110 

11 

121 

11 

132 

12 

96 

12 

108 

12 

120 

12 

132 

12 

144 

The  multiplication  table  promiscuously  arranged  without 
the  products  : 

Multiplicands      12      9    11      8     10      7      4      6      3      5      2 
Multipliers         !?1110_9_8_7_6_5_4_3J2 

ANOTHER   FORM. 

Multiplies     47    10      258    11      0369    12 

Multipliers  J*    _j?_?_2_2_?_§_?_?J?_J5_? 

147    10      258    11      0369    12 


147  10   258  11   0369  12 


147  10   258  11   0369  12 


147  10   258  11   0369  12 


147  10   258  11   0369  12 


147  10   258  11   0369  12 
8989898989898 


30  MULTIPLICATION. 

Multpl'dl      47    10      258    11      0369    12 
Mulp'rs_9    _8_9_89_89898989 

147    10      258    11      0369    12 


147    10      258    11      0369     12 
H  ^v  j^    11    12    10  jl   JiJ    10    11    12    10    11 

147    10      258    11      0369    12 

1?  19  1^  1?  i?  Ji  1?  1?  11  J^  J!^  \H  J3 

[Let  pupils  write  the  above  table  and  the  respective  products  on 
their  slates  ;  also  recite  the  products  without  being  written  till  they 
can  do  so  promptly  without  mistakes. 

Art.  10.  —  Multiplication  by  one  figure. 

MENTAL  EXERCISES. 

At  3  cents  each,  how  much  will  2  oranges  cost  ?  4?  6?  3?  5? 

Answer.  —  If  1  orange  cost  3  cents,  2  oranges  will  cost  2  times 
or  twice  3  cents,  which  are  6  cents. 

At  4  cents  each,  how  much  will  3  lemons  cost  ?  5  ?  2  ?  4  ? 
6  ?  3  ?  5  ? 

At  5  cents  each,  how  much  will  4  melons  cost  ?  3  ?  5  ?  2  ? 
4?  6? 

At  6  cents  each,  how  much  will  5  pine  apples  cost  ?  3  ?  2  ? 
4?  6?  5? 

At  7  cents  a  pint,  how  much  will  2  pints  of  cherries  cost  ? 
4?  6?  3?  5? 

At  8  cents  a  pint,  how  much  will  3  pints  of  strawberries  cost  ? 
5?  2?  4?  6? 

At  9  cents  a  quart,  how  much  will  4  quarts  of  chestnuts  cost  ? 
2?  6?  3?  5? 

At  10  cents  a  pound,  how  much  will  5  pounds  of  sugar  cost  ? 
3?  2?  4?  6? 

At  11  cents  a  pound,  how  much  will  6  pounds  of  cheese 
cost?  4?  3?  5?  2? 

At  12  cents  a  pound,  how  much  will  3  pounds  of  ginger  cost  ? 
5?  2?  4?  6? 


MULTIPLICATION. 


31 


EXAMPLES  FOE  THE   SLATE. 

EXAMPLE  1.— Multiply  4326  by  5. 

Process.— 5  times  6  are  30  ;  write  0  units  and  )  (  4326 

carry  the  three  tens  as  in  addition  ;  5  times  2  >•  written  -|  ^ 

tens  are  10  tens,  and  three  carried  are  13  tens  ; )  (  Ans.  21630 

write  the  3  tens  and  carry  the  1.     Thus  proceed. 

Proof. — 4  times-}-l  time  the  number  are  5  times  the 
number.  Therefore  multiply  by  4,  and  ta  the  product 
add  the  multiplicand  ;  the  result  will  be  same  as  before 
if  correct.  Or  repeat  the  process,  inverting  the  order 
of  the  figures,  as  6  times  4,  2  times  4,  &c. 


PBOOF. 
4326 
4 

17304 
21630 


RULE. —  Write  the  multiplier  under  the  right-hand  figure 
of  the  multiplicand,  and  draw  a  line  under  it.  Begin  at  the 
right  hand  and  multiply  each  figure  of  the  multiplicand  by 
the  multiplier,  carrying  as  in  addition. 

PROOF. — Multiply  by  a  number  1  less  than  before,  and  to 
the  product  add  the  multiplicand.  If  the  result  is  the  samet 
it  is  correct. 


(20 

"Write 

(3.) 
(4.) 
(5.) 
(6.) 
(7.) 
(8.) 
(9.) 
(10.) 

135024 

2 

multiply 

(11.) 
(12.) 
(13.) 
(14.) 
(15.) 
(16.) 
(17.) 
(18.) 

135024 
3 

by  2, 

(19.) 
(20.) 
(21.) 
(22.) 
(23.) 
(24.) 
(25.) 
(26.) 

135024 
4 

thus  and 

246135 
350246 
461350 
502461 
613502 
153042 
264153 
305264 

each  number 
416305 
520416 
631502 
310542 
421653 
532064 
643105 
520641 

3,  and  4 

165320 
510342 
621435 
205614 
136205 
103245 
124356 
235460 

Multiply  by  5  and  6  : 


(27.) 
(28.) 
(29.) 

(30.) 
(31.) 
(32.) 


679867 

579689 
497867 
938796 
479685 
596879 


(33.)  689769 

(34.)  786975 

(35.)  879789 

(36.)  787980 

(37.)  979899 

(38.)  676869 


(39.)  486979 
(40.)  579687 
397869 
278697 
697987 


(41.) 

(42.; 
(43.; 


(44.)  708090 


32  MULTIPLICATION. 

MENTAL   EXEECISES. 

At  2  shillings  a  yard,  how  mucli  will  7  yards  of  calico  cost  ? 
8?  9?  5? 

At  3  shillings  a  yard,  how  much  will  8  yards  of  ribbon  cost  ? 
7?  9?  6? 

At  4  shillings  a  yard,  how  much  will  9  yards  of  silk  cost  ? 
7?  4?  8? 

At  5  dollars  a  yard,  how  much  will  7  yards  of  cloth  cost  ? 
9?  6?  8? 

At  6  cents  a  skein,  how  much  will  8  skeins  of  sewing  silk 
cost?  6?  9?  7?  5? 

At  7  shillings  a  pound,  how  much  will  9  pounds  of  wool 
cost?  7?  8?  4? 

At  8  dollars  each,  how  much  will  7  coats  cost  ?  9  ?  6  ? 
8?  5? 

At  9  cents  a  yard,  how  much  will  8  yards  of  muslin  cost  ? 
8?  7?  9?  4V 

At  10  cents  a  spool,  how  much  will  9  spools  of  cotton  cost  ? 
7?  5?  8?  6? 

At  11  dollars  each,  how  much  will  7  shawls  cost  ?  5  ?  3  ? 
9?  8? 

At  12  shillings  a  pair,  how  much  will  8  pairs  of  gloves  cost  ? 
6?  9?  7?  4? 

EXAMPLES  FOB  THE   SLATE. 

(45.)     14725803                          14725803                          14725803 
7  8  9 

Write  thus  and  multiply  each  number  by  7,  8,  and  9  : 
(46.)     69142580  (50.)     17452083  (54.)     12740853 

(47.)     36914758  (51.)     61794250  (55.)     20859631 

(48.)     69147280  (52.)     39641278  (56.)     31962745 

(49.)     47258069  (53.)     50863194  (57.)     45270836 

Art,  11.— Multiplication  by  two  or  more  figures. 

MENTAL   EXEECISES. 

At  2  dollars  each,  how  much  will  10  caps  cost  ?  12  ?  11  ? 
9? 

At  3  dollars  each,  how  much  will  11  hats  cost  ?  10  ?  12  ?  8  ? 


MULTIPLICATION.  33 

At  4  dollars  each,  how  much  will  12  bonnets  cost  ?  10  ? 
7?  11? 

At  5  shillings  a  pair,  how  much  will  10  pairs  of  socks  cost  ? 
12  ?  9  ?  11  ? 

At  6  shillings  a  yard,  how  much  will  11  yards  of  linen  cost  ? 
9?  12?  10? 

At  7  cents  each,  how  much  will  12  papers  of  pins  cost  ?  10  ? 
8?  11? 

At  8  cents  each,  how  much  will  10  papers  of  needles  cost  ? 
12?  8?  11? 

At  9  cents  a  dozen,  how  much  will  11  dozen  of  buttons  cost  ? 
9?  12?  10? 

At  10  cents  a  yard,  how  much  will  12  yards  of  cambric  cost  ? 
10?  7?  11? 

At  11  dollars  each,  how  much  will  11  shawls  cost  ?  8  ?  12  ? 
10? 

At  12  dollars  each,  how  much  will  12  pieces  of  oilcloth  cost  ? 
10?  6?  11?  9?  8?  7? 

EXAMPLES   FOB   THE   SLATE. 

EXAMPLE  58.—  Multiply  172829  by  12. 

1728°9 
Process.  --The  same  u:;  multiplying  by  units  or   one  TO 

figure;  12  times  9,  *o. 


Write  thus  and  multiply  each  number  by  10,  11,  and  12  : 
(59.)     96303527  (62.)     30S52741  (65.)     30691472 

(60.)     85274196  (63.)     27419630  (66.)     25803691 

(61.)     74196308  (64.)     19630852  (67.)     50642839 

Art.  12,  —  Multiplication  by  numbers  greater  than  12. 

EXAMPLE  68.—  Multiply  28357  by  234. 

Process.—  Multiplying  by  234  is  the  same  as  multi-  28357 

plying  by  2  hundreds,  3  tens  and  4.     4  times  28357  234 

=113428    by  the  same  process  as  before.     3  times  113428 

28357—85071,  and  since  the  multiplier  is  tens  the  pro-  85071 

duct  will  be  tens  ;  therefore  write  the  right  hand  figure  56714 

under  tens.     For  a  similar  reason  write  the  right  hand 
figure  of  the  product  of   28357X2  under  hundreds.    Ans-  « 
Then  the  products  of  the  number  multiplied  by  2  hundreds,  3  tens 
and  4  added  together  will  be  its  product  multiplied  by  234. 
2* 


MULTIPLICATION. 


BULK — Write  the  multiplier  under  the  multiplicand. 
Begin  at  the  right  hand,  and  multiply  by  each  figure  of  the 
multiplier  separately,  writing  under  each  the  right-hand 
figure  of  its  product :  with  the  other  figures  in  their  proper 
order. 

Add  the  products  of  each  figure  together,  and  the  product 
of  the  two  numbers  will  be  found.  • 

PROOF. — Eepeat  the  multiplication,  only  inverting  the  order 
of  the  figures,  as  in  the  above  example  ;  instead  of  4  times 
1  say  1  times  4. 

When  the  multiplier  contains  a  cipher  it  may  be  passed  over,  but 
the  right  hand  figure  of  the  next  product  must  be  written,  not  un- 
der the  cipher,  but  its  own  multiplier. 


(69.) 
(70.) 
(71.) 
(72.) 
(73.) 
(74.) 
(75.) 
(76.) 
(77.) 
(78.) 
(79.) 

144X13 
245X18 
356X24 
587X^5 
789X56 
890X67 
901X78 
234X89 

987X99 
876X98 
765X87 

(80.; 
(81.; 
(82.; 

(83; 


EXAMPLES. 

1426X31 
2536X42 

3675X54 
4879X65 
(84.)  5098X76 
(85.)  6109X87 
(86.)  7432X98 
(87.)  8987X89 
(88.)  9786X78 
(89.)  1067X99 


(90.)  15789X125 

(91.)  21478X234 

(92.)  34890X345 

(93.)  41256X456 

(94.)  54675X567 

(95.)  67812X678 

(96.)  78569X789 

(97.)  86453X897 

(98.)  95387X901 

(99.)  92896X802 


Special  Rules. 

Art.  13. — Multiplication  of  numbers  having  ciphers  on 
the  right  hand. 

Ex.  100.— Multiply  245  by  100. 


Process.— Write  245  with  two  ciphers  (24500)  on  the 
right  hand.  This  changes  the  local  value  of  the  figures 
the  same  as  multiplying  by  100. 


245 

100 

000 
000 
245 

Ans.  24500 


MULTIPLICATION. 


35 


Ex.  101.— Multiply  256  by  300. 

Process. — 3  times  256  are  768,  and  two  ciphers  are  to 
"be  written  after  this  product  for  the  same  reason  as 
before. 

Ex.  102.— Multiply  4600  by  32000. 

Process. — 46X32=1472.  Write  five  ciphers  at  the 
right  hand,  for  the  product  of  hundreds  by  thousands 
is  hundreds  of  thousands  by  the  general  rule, 


256 
300 

Ans.  76800 


4600 
32000 


9200000 
138  

Ans.  147200000 

RULE. —  When  the  multiplier  is  10,  100,  1000,  &c.,  write 
as  many  ciphers  as  it  contains  on  the  right  hand  of  the  mul- 
tiplicand. 

In  other  cases,  write  and  multiply  the  other  figures  as  if 
they  had  no  ciphers  on  the  right  hand,  and  annex  to  the 
product  as  many  ciphers  as  were  not  used  in  both  numbers. 


EXAMPLES. 

(103.) 

2250X  10 

(109.) 

1020000X  500 

(104.) 

3500X1200 

(110.) 

276000X    3 

(105.) 

4732X1000 

(111.) 

375X30100 

(106.) 

130X  51 

(112.) 

17020X  1000 

(107.) 

356X  100 

(113.) 

1000000x10000 

(108.) 

7000X  700 

(114.) 

1004000X10500 

MULTIPLICATION    BY    COMPOSITE   NUMBEKS. 

Art.  14. — A  Composite  number  is  one  which  is  the  pro- 
duct of  two  other  numbers  ;  as,  30  composed  of  6  X  5. 

Ex.  115.— Multiply  29  by  24. 


Process.— Since  4  times  6=24,  4  tunes  6  times  29=24 
times  29. 


29 
6 

174 
4 


RULE. — Find  two  or  more  numbers  which  being  multiplied 
together  will  produce  the  given  multiplier.      Multiply   the 


36 


MULTIPLICATION. 


multiplicand  by  one  of  them,  and  its  product  by  another,  till 
all  the  factors  are  used. 


(116.)  115X15 

(117.)  123X16 

(118.)  39X36 

(119.)  162X18 


EXAMPIiES. 

(120.)  126X27 
(121.)  99X32 
(122.)  265X35 


(123.)  324X48 
(124,)  87X63 
(125.)  405X108 


(126.) 

3456789X  1019 

(135.)  871496X2468 

(127.) 

9830291X  7305 

(136.)  397684X6005 

(128.) 

5006284X  6635 

(137.)  469537X3708 

(129.) 

4000059X  7239 

(138.)  873576X8764 

(130.) 

873000X  1000 

(139.)  468937X7056 

(131.) 

257000X  4000 

(140.)  798600X8750 

(132.) 

5749362X  3827 

(141.)  750000X9000 

(133.) 

4327000X  3500 

(142.)  596875X9678 

(134.) 

13786926X85043 

(143.)  658907X7869 

(144.)  963852 

7  1  4  —  Multiplicand. 

475180 

6  3  9—  Multiplier. 

145.  Multiply  seventy-six  thousand  by  sixty-eight  hundred 
and  four. 

146.  Multiply  nine  million  and  eight  thousand  by  five  hun- 
dred thousand  and  sixty. 

147.  Multiply  eighty-seven  thousand,  six  hundred  and  three, 
by  nine  thousand,  eight  hundred  and  sixty-five. 

148.  Multiply  eighty-three  thousand,  four  hundred  and  fifty 
seven,  by  six  thousand,  eight  hundred  and  thirty-five. 

149.  Multiply  nine  hundred  and  four  thousand,  by  ten  thou- 
sand and  two  hundred. 

150.  Multiply  eighty  thousand  and  six  hundred,  by  seven 
thousand  and  two. 

151.  Multiply  three  million,  two  hundred  and  forty  thou- 
sand, by  three  hundred  and  twenty-four  thousand. 

152.  Multiply  three  hundred  and  four  thousand  and  seven 
hundred,  by  ninety-seven  thousand,  six  hundred  and  three. 


MULTIPLICATION.  37 

153.  Multiply  eight  million,  six  hundred   and  forty-three 
thousand,  by  nine  thousand,  two  hundred  and  thirty. 

PBACTICAIi  EXAMPLES. 

154.  At  4  dollars  a  yard,  how  much  will  25  yards  of  cloth 
cost  ? 

155.  How  many  yards  in  seven  pieces  of   muslin  if  each 
piece  contain  28  yards  ? 

156.  At  9  dollars  a  barrel,  how  much  will  124  barrels  of  flour 
cost  ? 

157.  At  112  dollars  an  acre,  how  much  will  a  farm  contain- 
ing 270  acres  cost  ? 

158.  If  a  man  travel  32  miles  a  day,  how  far  will  he  travel 
in  24  days. 

159.  At  50  cents  a  bushel,  how  much  will  136  bushels  of 
apples  cost  ? 

160.  At  100  cents  a  day,  how  much  will  a  laborer  earn   in 
110  days  ? 

161.  At  12   cents  a  pound,  how  much  will  120  pounds  of 
beef  cost? 

162.  At  $97  each,  how  much  will  15  horses  cost  ? 

163.  At  $10  a  barrel,  how  much  will  225  barrels  of  flour 
cost? 

164.  At  $56  a  head,  how  much  will  25  cows  cost  ? 

165.  At  $23  an  acre,  how  much  will  99  acres  of  land  cost  ? 

166.  In  an  orchard  there  are  84  rows  of  trees  and  63  trees  in 
each  row ;  how  many  trees  are  there  ? 

167.  In  a  bale  of  sheeting  there  are  22  pieces,  and  in  each 
piece  27  yards  ;  how  many  yards  in  all  ? 

168.  In  a  hogshead  there  are  63  gallons  ;    how  many  gallons 
in  23  hhds. 

169.  In  a  box  of  calico  there  are  94  pieces,  and  each  piece 
contains  35  yards ;  how  many  yards  in  all  ? 

170.  At  $17  a  barrel,  how  much  will  211  barrels  of  molasses 
cost. 

Many  more  such  examples  will  be  found  among  the  Promiscuous 
Examples,  after  Division,  and  U.  S.  Money. 

? 


UNIVERSITY    I 


38  DIVISION. 


DIVISION. 

Art.  15. — Division  is  finding  either  how  often  one  number 
is  contained  in  another  ;  or  one  of  as  many  equal  parts  of 
a  number  as  are  expressed  by  another  number  ;  thus, 

We  find  either  that  5  is  contained  in  30  6  times ;  or  that  6  is 
one  of  5  equal  parts  of  30. 

The  number  to  be  divided  is  called  the  Dividend;  the 
dividing  number,  the  Divisor ;  the  number  found  or  the 
answer,  the  Quotient  J  and  that  part  of  the  dividend  less 
than  the  divisor,  which  is  sometimes  left  after  division,  is 
called  the  Remainder. 

The  sign  of  division  is  (-£-)  a  horizontal  line  between 
two  dots,  written  after  the  dividend  and  before  the 
divisor. 

Division  may  also  be  expressed  by  writing  the  divisor 
under  the  dividend.  In  this  way  the  remainder  becomes 
a  part  of  the  quotient  at  the  right  hand.  Such  an  ex- 
pression is  called  a  fraction;  as  |  (read)  one-half;  £, 
one-third ;  f ,  two-thirds  ;  J,  three  fourths  or  quarters  ; 
f,  four-fifths,  &c.,  &c. 

ILLUSTRATION.—  Divide  27  by  4.  27-H=  or  *£-  =6  and  3  left,  or  6|. 
27  is  the  dividend,  4  the  divisor,  6|  the  quotient,  or  6  the  quotient 
and  3  the  remainder. 

KEMABK  1. — Dividing  a  number  by  2,  divides  it  into  2  equal  parts 
each  of  which  is  called  one-half  (written  5;)  dividing  it  by  3  divides 
it  into  3  equal  parts,  each  called  one-third  (-£. )  So  dividing  by  4 
gives  one-fourth  (£,)  by  5  one-fifth  (£,)  by  6,  7,  10,  15,  &c.,  gives' 
one-sixth  ( £, )  one-seventh  ( |,  )  one-tenth  (-fa, )  one  fifteenth 
(iV)  &c. 

2.  The  divisor  and  quotient  correspond  with  the  factors  in  multi- 
plication, and  the  dividend  with  the  product.     Hence 

3.  When  one  factor  and  the  product  is  given,  the  other  factor  may 
be  found  by  division. 

4.  One  of  the  factors  must  be  of  the  same  name  as  the  dividend, 
and  one  of  as  many  equal  parts  of  it  as  are  expressed  by  the  other. 
If  the  divisor  is  one  of  the  equal  parts,  the  quotient  is  the  number  of 


DIVISION.  39 

parts  ;  but  if  the  divisor  is  the  number  of  parts,  the  quotient  is  one 
of  the  parts  ;  thus, 

20  cts.  -r-  5  cts.  =  4  (equal  parts);  but  20  cts.  ~  4  (equal  parts) 
=  5  cts.  (one  of  the  equal  parts  of  20). 

Art.  16. — There  are  two  methods  of  division,  Short  and 
Long.  In  short  division  the  process  is  mental,  and  only 
the  result  written.  This  is  used  when  the  divisor  is  less 
than  12.  In  long  division  the  process  and  result  are 
both  written. 

In  dividing  the  divisor  is  usually  \mtten  before  the  dm-  3)15 
dend,  and  the  quotient  in  short  division  under  it,  thus,  5 

In  long  division  after  it,  thus :  13)|| (2 


Short  Division. 

DIVISION  TABLE  WITHOUT   THE  QUOTIENTS. 

3)     3      69121518212427      30^33^  36 

4)481216^20^242832364044  48^ 

5)510^52025^30^35^40^45^5055  60 

6<J?12JIS^<W^^^546066  Z?. 

^Zlij^^^^^^6iZ5.11  §4 

8;.    ^    16    24    32    40    48    56^    64      72      80      88^  96 

9)j?18_273645546372      8^     90      99^  108_ 

10)^.020^040^50607080      90^    100      110  120 

11)  _11    22^  ^3    44    55    66^    11_    88^     99^    110      121  132 

12)  12  24  36  48  60  72  84  96  108  120   132  144 


40  DIVISION. 

The  same  table  promiscuously  arranged  : 
2)4101622281420      61218^4 

3)  6      15      2£    33     3    12^     21^     30      9      18      27  36 

4)  _8     20?2_Mi!6_??.4).!!2_?i_3?  48 
5)^02540555203550^15^30^5  60 
6)^230486662442^60183654  72 

7)l^^^Zll^l?_™?I:t?.J5    ?f 

8)^6406488832^51680244872  96 

9)JL8     ^5.     Z?.     99     9    36     ^>3     ^0^7548^  108 

10)  ^0      50     _80    100  JA)    40     70    WO   _30     60     90  120 

11)  ^2      55.88121114477110336699  132 

12)  _24  60  ^  1§2  12_  ±8  8£  120  36  72108  144 

MENTAL   EXERCISES. 

At  2  cents  each,  how  many  peaches  can  be  bought  for  8 
cents?  4?  12?  16?  20?  24?  28?  6?  10?  14?  18? 
24? 

Process.— If  2  cents  will  buy  1  peach,  8  cents  will  buy  as  many 
peaches  as  there  are  times  2  cts.  in  8  cents,  which  are  4. 

Or  if  2  cents  will  buy  1  peach,  half  as  many  as  the  cents,  8  cents  will 
buy  half  of  8  peaches,  which  are  4. 

At  3  cents  each,  how  many  oranges  can  be  bought  for  6  cts  ? 
12?  18?  24?  30?  36?  9?  15? 

At  4  cents  each,  how  many  melons  can  be  bought  for  12 
cents?  8?  16?  24?  32?  40?  48?  20?  28?  36?  44? 

At  5  shillings  a  pound,  how  many  pounds  of  tea  can  be 
bought  for  15  shillings  ?  25?  35?  20?  30?  45?  55? 
65?  40?  50?  60? 


EXAMPLES.  41 

At  6  shillings  a  yard,  how  many  yards  of  linen  can  be  bought 
for  12  shillings  ?  24?  36?  48?  60?  18?  30?  42?  66? 

72? 

Divide  the  following  numbers,  and  write  the  remain- 
ders over  the  divisors  at  the  right  hand  of  the  quo- 
tient ;  as,  6)38 

~6f 

Let  the  pupils  also  recite  them  without  seeing  the 
answers  written : 

2)511  1723  391521  7131925 
3)  8_  17  2735  514233211202837 
4)11^21^3645717314113263749 
5)  13_  27_  44  J38  8  22  j*7  »3  H  23^  48  63 
6)1431  50  70  92744  62  20  25  _56  J4 
7)  17_  37  60j8irL29537423^  i(37  _89 

8)1945  70     93133559  852735     75  101 

9)2150  80    103    15    40   J55  j)7   JJ1   J59      84  112 

10)  23    55  89    113    17^   43    73  104    34    43     93  JL25 

11)_25  _61  _90  _127  JL9  _47  _78  113  _36_48_jLOO  ^31 

12)  27  JO  100  13(5.?!_5^j3?  122  38  53  112  150 

* 

MENTAL  EXEBCISES. 

At  7  cents  a  quart,  how  many  quarts  of  milk  can  be  bought 
for  14  cents?  28?  35?  42?  56?  49?  63?  21?  70? 

At  8  shillings  a  bushel,  how  many  bushels  of  corn  can  be 
bought  for  24  shillings  ?  40?  16?  32?  48?  64?  80?  56? 


42  DIVISION. 

At  9  dollars  a  barrel,  how  many  barrels  of  flour  can  be 
bought  for  18  dollars  ?  36?  27?  45?  63?  54?  72? 
99?  81? 

At  10  shillings  a  bushel,  how  many  bushels  of  wheat  can  be 
bought  for  30  shillings  ?  20?  50?  40?  70?  60?  90? 
80?  100? 

At  11  cents  a  pound,  how  many  pounds  of  rice  can  be  bought 
for  33  cents  ?  22  ?  44  ?  66  ?  55  ?  77  ?  99  ?  88  ?  110  ? 

At  12  cents  a  pound,  how  many  pounds  of  coffee  can  be 
bought  for  24  cents  ?  48?  60?  36?  72?  96?  84?  120? 

If  three  apples  cost  6  cents,  how  much  will  one  apple  cost  ? 

Process.  — If  three  apples  cost  6  cents  1  apple  will  cost  as  many 
cents  as  3  is  contained  times  in  6  ;  or  i  of  6  cents,  which  is  2  cents. 

If  5  peaches  cost  15  cts.,  how  much  will  1  peach  cost  ? 

If  7  melons  cost  42  cts.,  how  much  will  1  melon  cost  ? 

If  9  bushels  of  chestnuts  cost  36  dollars,  how  much  will  1 
bushel  cost  ? 

If  8  barrels  of  flour  cost  72  dollars,  how  much  will  one  bbl. 
cost? 

If  12  yards  of  cloth  cost  84  dollars,  how  much  will  1  yard 
oost? 

EXAMPLES   FOB   THE   SLATE. 

EXAMPLE  1.— Divide  4325  by  5. 

Process. — Since  it  is  not  easy  to  divide  so  large 


a  number  at  once,  it  is  considered  as  separated 
into  parts  ;  as  4000+300+20+5. 

If  5  were  contained  in  4,  the  first  figure  at  the 
left  of  the  quotient  would  be  thousands  ;  but  5  is 
not  contained  in  4,  therefore  add  4000  to  300  and  - 
divide  the  hundreds.  5  is  contained  in  43  hun- 
dred (4300)  8  hundred  (800)  times,  and  3  hun- 
dred (300)  remainder.  Add  the  remainder  to  the 
2  tens  (20)  and  divide  the  tens.  5  is  contained 
in  32  tens  (320)  6  tens  times  and  2  tens  remain- 


300 

20 

5 


•  Parts  of 
the  divi- 
dend. 


800 )  Parts   of 
60  V  the  quo- 
5  )  tient. 
Comm'nly  written 

-    5)4325 
Ans.  "865 


der,  which  add  to  5  and  divide.     The  last  part  of 

the  quotient  will  be  5.     The  sum  of  all  the  parts  is  the  quotient  or 

answer  required. 

The  same  result  is  obtained  by  the  following  process,  commonly 
used  : 

5  into  43—8  and  3  remaining.     Write  the  8  under  the  last  figure 


EXAMPLES. 


43 


divided,  and  prefixing  the  3  remaining  to  the  2,  divide  5  into  32—6 
and  2  remaining.  Write  the  6  under  the  last  figure  divided,  and  thus 
proceed. 

RULE  FOR  SHORT  DIVISION. —  Write  the  divivor  at  the  left 
of  the  dividend,  and  draw  a  line  under  the  latter. 

.Begin  at  the  left,  and  divide  each  figure  separately.  Write 
each  figure  of  the  quotient  under  the  last  figure  divided,  and 
prefix  the  remainder,  if  there  is  any,  to  the  next  figure  of 
the  dividend  and  divide  again. 

If  there  is  a  remainder  of  ter  dividing  all  the  figures  of  the 
dividend,  write  it,  with  the  divisor  under  it,  at  the  right  hand 
of  the  quotient. 

If  there  is  no  other  figure,  a  cipher  must  be  written  in  any  place 
except  at  the  left  hand. 

PROOF. — Multiply  the  divisor  by  the  quotient  and  add  the 
remainder.  The  product  should  be  like  the  dividend. 


(2.)     2)135024 


3)135024 


4)135024 


Write  thus,  and  divide  each  number  by  2,  3,  and  4. 


(3.)  246135 

(4.)  350246 

(5.)  461350 

(6.)  502461 

(7.)  613502 

(8.)  153042 

(9.)  264153 
305264 


(10. 


(11.) 

(12.) 
(13.) 
(14.) 
(15.) 
(16.) 
(17.) 
(18.) 


416305 
520416 
631502 
310542 
421653 
532064 
643105 
520641 


(19.)  165320 

(20.)  510342 

(21.)  621435 

(22.)  205614 

(23.)  136205 

(24.)  103245 

(25.)  124356 

(26.)  235460 


Divide  each  number  by  5  and  6. 


(27.) 
(28.) 
(29.) 
(30.) 
(31.) 
(32.) 


679867 
579689 
497867 
938796 
479685 
596879 


(33.)  689769 

(34.)  786975 

(35.)  879789 

(36.)  787980 

(37.)  979899 

(38.)  676869 


(39.) 
(40.) 
(41.) 
(42.) 
(43.) 
(44.) 


486979 
579687 
397869 
278697 
697987 
708090 


DIVISION. 


MENTAL  EXERCISES. 

If  6  pounds  of  sugar  cost  75  cents,  how  much  will  1  pound 
cost? 

At  7  cents  a  pound  how  many  pounds  of  fish  can  be  bought 
for  25  cts.  ?  31  cts.  ?  37  cts.  ?  45  cts.  ?  65  cts.  ?  72  cts.  ?  90  cts.  ? 

At  8  dollars  a  barrel,  how  many  barrels  of  flour  can  be  bought 
for  $28  ?  $18  ?  $35  ?  $50  ?  $42  ?  $60  ?  $75  ?  $83  ?  $90  ?  $100  ? 

If  12  pounds  of  rice  cost  100  cts.,  how  much  will  1  Ib.  cost  ? 

At  12  cents  a  pound,  how  many  pounds  of  starch  can  be 
bought  for  30  cts.  ?  40  cts.  ?  50  cts.  ?  64  cts.  ?  70  cts.  ?  80  cts.  ? 
90  cts.?  100  cts.?  110  cts.?  125  cts.?  150  cts.? 

At  9  dollars  a  ton,  how  many  tons  of  coal  can  be  bought  for 
$100  ?  $75  ?  $67  ?  $55  ?  $48  ?  $42  ?  $37  ?  $35  ?  $94  ? 

If  11  tons  of  hay  cost  125  dollars,  what  will  1  ton  cost. 

At  10  dollars  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $34  ?  $45  ?  $56  ?  $64  ?  $75  ?  $84  ?  $96  ?  $125  ? 


EXAMPLES  FOE  THE  SLATE. 

(45.)  7)14725803     8)14725803 
Write  thus,  and  divide  by  7,  8,  and  9. 

69142580 
36914758 
69147280 
47258069 
58036914 
72580369 
80369147 


9)14725803 


(46.) 
(47.) 
(48.) 
(49.) 
(50.) 
(51.) 
(52.) 
(53.) 


(54.) 
(55.) 
(56.) 
(57.) 
(58.) 


91472580 


(61.; 


17452083 
61794250 
39641278 
50863194 
75208639 
42785301 
83096417 
94127528 


(62.) 
(63.) 

(64.) 
(65.) 
(66.) 
(67.) 
(68,) 
(69.) 


12740853 
20859631 
31962745 
45270836 
53081964 
64195278 
78526309 
86304190 


Divide  the  following  numbers  by  10,  11,  and  12. 


(70.) 
(71.) 
(72.) 
(73.) 
(74.) 
(75.) 


96308527 
85274196 
74196308 
63085274 
52741963 
41963085 


(76.) 
(77.) 
(78.) 
(79.) 
(80.) 
(81.) 


30852741 
27419630 
19630852 
93824605 
82493057 
71938246 


(82.) 
(83.) 
(84.) 
(85.) 
(86.) 
(87.) 


30691472 
25803691 
50642839 
75039428 
64283917 
14725803 


DIVISION.  45 


Art,  17.— Long  Division, 

EXAMPLE  1.  Divide  97836  by  18. 

Process.— The  same  as  in  short  division,  ex-  Divsr.Divid'd.Quoti'nt 
cept  the  quotient  is  placed  at  the  right  hand,  and      18)97386(5435^ 
the  products,  with  the  remainders,  are  written  90 

under  the  parts  of  the  dividend  used.  ~78 

Quotient  figures  must  often  be  found  by  trial,  72 

and  if  the  product  of  any  trial  figure  and  the  di-  -go 

visor,  is  greater  than  the  part  of  the  dividend 

used,  that  figure  is  too  great ;  on  the  other  hand,  

if  the  remainder  is  greater  than  the  divisor,  the 
quotient  figure  is  too  small. 

Very  often  a  quotient  figure  may  be  found,  or  6 

nearly  so,  by  dividing  the  first  figure  or  two  in 
the  dividend,  by  the  first  figure  in  the  divisor,  allowing  more  or  less 
for  carrying. 

RULE. —  Write  the  divisor  at  the  left  of  the  dividend,  and 
leave  a  place  for  the  quotient  on  the  right. 

Find  how  many  times  the  divisor  is  contained  in  the  few- 
est left-hand  figures  of  the  dividend  that  contain  it,  and  write 
the  number  in  the  quotient. 

Multiply  the  divisor  by  this  quotient  figure  and  write  the 
product  under  that  part  of  the  dividend  which  was  found  to 
contain  the  divisor.  Subtract  the  product  from  the  figures 
above  it ;  to  the  remainder  bring  down  the  next  figure  in  the 
dividend,  and  divide  again. 

Continue  dividing  till  all  the  figures  of  the  dividend  are 
used,  and  if,  after  bringing  down  a  figure,  the  number  thus 
formed  is  less  than  the  divisor,  write  a  cipher  (0)  in  the  quo- 
tient and  bring  down  another  figure. 

PBOOF. — The  same  as  in  short  division. 

EXAMPLES. 

(2.)  (3.)  (4.) 

13)H7256(  U)152768(  15)175265( 


DIVISION. 


Divide— 

(5.)  3540  by  12    (17.)  5258  by  22 

(29.)  158500  by  50 

(6.)  8764  "  13    (18.)  10713  "  14 

(30.)  140998  "  26 

(7.)  5200  "  15    (19.)  11829  "  18 

(31.)  116873  "  28 

(8.)  5499  "  13    (20.)  11347  "  19 

(32.)  127364  "  68 

(9.)  7556  "  16    (21.)  39825  "  27 

(33.)  295482  "  74 

(10.)  4935  "  15    (22.)  48440  "  28 

(34.)  316704  "  96 

(11.)  9454  "  18    (23.)  37430  "  29 

(35.)  190850  "  25 

(12.)  6567  "  17    (24.)  13125  "  35 

(86.)  151445  "  35 

(13.)  5472  "16    (25.)  30825  "  45 

(37.)  135050  "  37 

(14.)  6614  "  19    (26.)  32806  "  47 

(38.)   55168  "  64 

(15.)  7348  "  17    (27.)  19608  "  43 

(39.)  5748435  "  63 

(16.)  9182  "  21    (28.)  41088  "  52 

(40.)  6480752  "  96 

(41.)  215045924  by  86     (44.) 

562752060  by  64 

(42.)  405147456  "  56     (45.) 

7254231  "  82 

(43.)  459932616  "  66     (46.) 

1053990  "  63 

Art,  18,— Special  Rules. 

Division  by  numbers  with  ciphers  on  the  right  hand. 
Ex.  47.— Divide  4720  by  100. 

Process.— Cut  off  two  figures  from  the  right  1,00)47,20=47^ 
hand  of  the  dividend  ;  since  this  changes  the 
value  of  the  rest,  the  same  as  dividing  by  100,  20  is  the  remainder 
unless  regarded  as  a  decimal  fraction. 

Ex.  48.— Divide  76900  by  300. 

Process.— The  same  as  before,  but  divide  also  by    3,00)769,00 
3,  the  other  factor  in  the  divisor.  055111(1 

3  0  0 

% 

KULE. — Cut  off  the  ciphers  on  the  right  of  the  divisor,  and 
and  as  many  figures  on  the  right  of  the  dividend.  Divide  the 
rest  of  the  dividend  by  the  rest  of  the  divisor,  if  greater  than 
one,  and  if  there  is  a  remainder,  annex  to  it'the  figures  cut  off 
from  the  dividend  for  the  true  remainder. 


DIVISION   BY  COMPOSITE   NUMBERS. 


EXAMPLES. 


Divide  — 

(49.) 

14100 

by 

600 

(53.) 

364000 

by 

6400 

(50.) 

365400 

« 

5000 

(54.) 

48000 

« 

1600 

(51.) 

138000 

« 

1000 

(55.) 

170000 

it 

400 

(52.) 

36009 

" 

1200 

(56.) 

1000009 

" 

300 

Art.  19. — Division  by  Composite  Numbers. 


Ex.  57.— Divide  1732  by  24. 

Process. — Dividing  by  3  divides  the  number  into 
one  of  three  equal  parts  and  1  remainder  ;  and  divid- 
ing this  by  8  divides  the  number  itself  into  one  of  3 
times  8  equal  parts,  and  one  remaining  to  each  of  the 
three  parts,  which  added  to  1,  the  first  remainder, 
makes  4  remainder. 


3)1732 

8)577+1 


72+4 


EULE. — Divide  by  one  of  the  factors,  and  the  quotient  thus 
found  by  the  other  factor.  If  there  are  remainders,  multiply 
the  second  one  by  the  first  factor,  and  add  the  first  remainder. 
Proceed  in  the  same  way  if  there  are  three  or  more  factors. 


EXAMPLES. 


Divide  — 

(58.) 

2976  by 

24 

(70.) 

186021  1 

>y  148 

(59.) 

134120  " 

56 

(71.) 

119753 

"  156 

(60.) 

155145  " 

54 

(72.) 

246813 

'  169 

(61.) 

105409  " 

63 

(73.) 

2500000 

"  200 

(62.) 

12148  " 

35 

(74.) 

1435792 

"  218 

(63.) 

1728  " 

144 

(75.) 

1579248 

'  227 

(64.) 

476345  " 

100 

(76.) 

1681357 

'  239 

(65.) 

567324  " 

111 

(77.) 

1792460 

4  244 

(66.) 

643142  " 

115 

(78.) 

1864219 

'  256 

(67.) 

792468  " 

125 

(79.) 

2004312 

'  260 

(68.) 

864219  " 

131 

(80.) 

3600000 

«  300 

(69.) 

975312  " 

142        (81.) 

1080000 

"  1200 

DIVISION. 


(82.) 

700239  by  123 

(90. 

)   470205  by  215 

(83.) 

1883187  " 

249 

(91. 

)   962984  " 

276 

(84.) 

593583  "  241 

(92. 

)   197776  " 

376 

(85.) 

1886826  " 

314 

(93. 

)   255136  " 

476 

(86.) 

478224  " 

324 

(94. 

)   124488  " 

342 

(87.) 

854661  " 

347 

(95. 

5698546  " 

829 

(88.) 

2539615  " 

439 

(96.)  5354320  " 

635 

(89.) 

1200000  " 

600 

(97.)  2400000  " 

800 

(98.) 

4874583 

by  643 

(109.) 

7046606  by 

898 

(99.) 

6079864 

"  719 

(110.) 

7977489  " 

923 

(100.) 

8264574 

"  846 

(111.) 

3769248  " 

948 

(101.) 

8095230 

"  935 

(112.) 

3779008  " 

548 

(102.) 

4674784 

"  694 

(113.) 

4991875  " 

625 

(103.) 

4663778 

"  1246 

(114.) 

9691836  " 

1234 

(104.) 

5332114 

"  1234 

(115.) 

5237479  " 

1823 

(105.) 

61142488 

"  4136 

(116.) 

18219071  " 

3001 

(106.) 

452491424 

"  3143 

(117.) 

70287492  " 

7117 

(107.) 

297396341 

"  3047 

(118.) 

16736642  " 

3497 

(108.) 

960000000 

"  8000      (119.) 

72000000  " 

9000 

(120.) 

3013974002  by 

3074 

(121.) 

25174363929  " 

30243 

(122.) 

881137279449  " 

90807 

(123.) 

153288487686  " 

407091 

^  (124.) 

49062139937803  "  7001009 

(125.) 

156000000000  " 

520000 

126.  Divide  one  hundred  and  twenty-seven  thousand,  by 
three  thousand,  seven  hundred  and  forty-six. 

127.  Divide  four  million,  six  hundred  and  sixty-three  thous- 
and, seven  hundred  and  seventy-eight,  by  three  thousand,  seven 
hundred  and  forty-three. 

128.  Divide  ten  million,   two  hundred  and  five   thousand, 
seven  hundred  and  twenty-one,  by  three  thousand  two  hun- 
dred and  forty-three. 


GENERAL  PP.INCIPLES  IN   DIVISION.  49 

129.  Divide  one  hundred  and  forty-one  thousand,  by  two 
thousand,  three  hundred  and  fifty. 

130.  Divide  eight  million,  eight  hundred  and  sixty  thou- 
sand, and  sixty,  by  one  thousand  and  thirty. 

131.  Divide  ninety-two   million   and  eighty  thousand,    by 
one  hundred  and  two. 

132.  Divide  twenty-three   million  and  forty  thousand,  by 
ninety-six  hundred. 

133.  Divide    two    million,    seven    hundred    and    thirty-six 
thousand,  three  hundred  and  seventy,  by  three  thousand  and 
seven. 

PRACTICAL   EXAMPLES. 

134.  If  it  would  take  1  man  3540  days  to  build  a  house,  how 
long  will  it  take  12  men  to  build  it  ? 

135.  At  $65  each,   how    many  cattle  can  be  bought  for 
$38740. 

136.  If  $38805  will  buy  597  cattle,  what  is  the  price  per 
head? 

137.  If  137  acres  of  land  cost  $17125,  what  is  the  price  per 
acre? 

138.  At  $125  an  acre,  how  many  acres  of  land  can  be  bought 
for  $17250. 

139.  At  $37  each,   how  many  cows  can    be    bought    for 
$14689. 

140.  If  396  cows  cost  $14652,  what  is  the  price  per  head. 

141.  At  $23  each,  how  many  coats  can  be  bought  for  $5451. 

142.  If  235  coats  cost  $5405,  what  is  the  cost  of  each  ? 

Many  more  such  examples  will  be  found  among  the  Promiscuous 
Examples. 


Art,  20,— General  Principles  in  Division. 

1.  Multiplying  the  dividend,  or  dividing  the  divisor  by 
any  number,  multiplies  the  quotient  by  that  number ; 
thus  24^-4=6.  (24X2)-r4=12,  or(6X2).  24 
=12. 


50  PROMISCUOUS  EXAMPLES. 

2.  Dividing  the  dividend,  or  multiplying  the  divisor, 
by  any  number  divides  the  quotient  by  that  number  ; 
thus,  24-r4=6  ;  ( 24-^2 )-r4=3,  or  (6-i-2);  24-^(4X2)= 
3,  or  (6-f-2.) 

3.  Multiplying  or  dividing  both  the  dividend  and  di- 
visor by  the  same  number,  does  not  alter  the  quotient ; 
thus,  24-f-4=6;   (24X2)-K±X2)=6  ; 

=6. 


Art,  21,— Promiscuous    Examples   in    Addition.    Sub- 
traction, Multiplication  and  Division. 

[Pupils  are  now  supposed  to  know  how  to  add,  subtract,  multiply 
and  divide.  The  following  examples  are  designed  to  teach  them 
when  to  apply  the  different  rules.  They  should  be  fully  explained  by 
the  pupils.  ] 

MENTAIj  EXEECISES. 

EXAMPUG  1. — A  boy  had  25  cents,  and  his  father  gave  him  25 
more  ;  how  many  did  he  then  have  ? 

Answer. — He  had  as  many  as  the  sum  of  25  cts.  added  to  25  cts., 
which  is  50  cts.  25+25=50. 

Ex.  2. — A  girl  had  50  cents,  and  paid  25  of  them  for  ribbon  ; 
how  many  had  she  left  ? 

Ans.—  She  had  as  many  as  the  difference  between  25  and  50  cents  ; 
or  as  the  remainder  after  subtracting  25  from  50  cts.,  which  is  25 
cts.  50—25=25. 

Ex.  3. — John  has  25  cents,  and  his  brother  has  3  times  as 
many  ;  how  many  has  his  brother  ? 

Ans. — He  has  as  many  us  3  times  25  cts.,  which  are  75  cents. 

25X3=75. 

Ex.  4. — Three  boys  have  75  cents  to  be  divided  equally  be- 
tween them  ;  how  many  will  each  boy  have  ? 

Ans. — Each  one  will  have  as  many  cents  as  there  are  times  3  in 
75,  or  4  of  75  cts.,  which  is  25  cts.  75+3=25. 


MENTAL  EXERCISES.  51 

Ex.  5.— If  75  cents  be  divided  equally  among  some  boys,  and 
each  one  receive  25  cents,  how  many  boys  will  there  be  ? 

Ans.—  There  will  be  as  many  boys  as  there  are  times  25  cts.  in  75 
cts.,  which  are  3  times.     Therefore  there  will  be  3  boys.     75-^-25=3. 

6.  At  4  cts.  each,  how  many  oranges  can  be  bought  for  32 
cents. 

7.  At  4  cents  each,  how  much  will  9  oranges  cost  ? 

8.  If  12  oranges  cost  48  cents,  what  is  the  price  of  each  ? 

9.  A  boy  picked  20  quarts  of   chestnuts  one  day,    and  14 
quarts  the  next ;  how  many  did  he  pick  in  the  two  days  ? 

10.  A  boy  had  20  quarts  of  chestnuts,  and  sold  14  quarts  of 
them  ;  how  many  had  lie  left  ? 

11.  At  9  cents  a  quart,  how  many  quarts  of  plums  can  be 
bought  for  72  cents  ? 

12.  At  9  cents  a  quart,  how  much  will  12  quarts  of  plums 
cost? 

13.  If  10  quarts  of  plums  cost  100  cents,  what  is  the  price 
per  quart  ? 

14.  James  has  agreed  to  pick  100  quarts  of  strawberries  in 
5  days  ;  how  many  must  he  pick  each  day  ? 

15.  James  has  agreed  to  pick  144  quarts  of  strawberries  ; 
how  long  will  it  take  him  if  he  pick  12  quarts  each  day  ? 

16.  James  picked  12  quarts  of  strawberries  a  day  ;  how  many 
quarts  did  he  pick  in  8  days  ? 

EXERCISE  I. — FOR  THE  SLATE   OR  BLACKBOARD. 

1.  A  farmer  has  327  sheep  in  one  flock,  and  258  in  another  ; 
how  many  has  he  in  both  ? 

2.  A  farmer  ha  I  640  lambs,  and  has  sold  325  ;  how  many 
has  he  left  ? 

3.  A  farmer  has  1950  bushels  of    oats,  and  can  carry  to 
market  75  bushels  at  a  load  ;  how  many  loads  will  there  be  ? 

4.  A  farmer  carried  to  market  25  loads  of  oats,  and  75  bush- 
els at  a  load  ;  how  many  did  he  carry  ? 

5.  A  farmer  has  1800  bushels  of  oats,  and  wishes  to  carry 


52  PEOMISCUOUS  EXAMPLES. 

them  all  to  market  in  24  loads  ;  how  many  bushels  must  he 
carry  each  time  ? 

6.  If  a  horse  eat  12  quarts  of  oats  a  day,  how  long  will  it 
take  him  to  eat  1728  quarts  ? 

7.  If  a  horse  eat  12  quarts  of  oats  a  day,  how  many  will  he 
eat  in  132  days  ? 

8.  If  a  horse  eat  1740  quarts  of  oats  in  145  days,  how  many 
will  he  average  each  day  ? 

9.  If  each  horse  eat  12  quarts  of  oats  a  day,  how  many 
horses  will  eat  1800  quarts  in  the  same  time  ? 

10.  If  100  horses  eat  1000  quarts  of  oats'  in  a  day,  how  many 
quarts  on  an  average  will  each  horse  eat  ? 

MENTAL   EXERCISES. 

1.  Henry  picked  36  quarts  of  cherries  and  his  brother  27 
quarts  ;  how  many  more  did  Henry  pick  than  his  brother  ? 

2.  At  8  cents  a  quart,  how  many  quarts  of  cherries  can  be 
bought  for  80  cents  ? 

3.  At  8  cents  a  quart,  how  much  will  12  quarts  of  cherries 
cost? 

4.  If  11  quarts  of  cherries  cost  99  cents,  what  is  the  price 
per  quart  ? 

5.  Henry  picked  29  quarts  of  cherries  and  his  brother  37, 
how  many  did  they  both  pick  ? 

6.  'At  18  cents  each,  how  many  knives  can  be  bought  for  72 
cents?  * 

7.  At  18  cents  each,  how  much  will  5  knives  cost  ? 

8.  If  6  knives  cost  90  cents,  what  is  the  price  of  each  ? 

EXERCISE  H. — FOB  THE   SLATE   OK  BLACKBOARD. 

11.  A  barrel  of  flour  contains  196  pounds,  how  many  pounds 
are  there  in  679  barrels  ? 

12.  How    many  barrels  of    the  same  will    contain    97412 
pounds  ? 

13.  If  there  are  97216  pounds  of  flour  in  496  barrels,  how 
many  pounds  are  there  in  each  ? 

14.  A  flour  dealer  bought  5624  barrels  of  flour,  and  has  since 
sold  3768  of  them,  how  many  has  he  left  ? 


MENTAL   EXERCISES.  53 

15.  At  $14  a  barrel,  how  many  barrels  of  flour  can  be  bought 
for  $11578  ? 

16.  At  $15  a  barrel,  how  much  will  3246  barrels  of  flour  cost  ? 

17.  If  45  barrels  of  flour  cost  $540,  what  is  the  price  per 
barrel  ? 

18.  If  a  ship  sail  96  miles  each  day,  how  long  will  it  take  her 
to  sail  2688  miles  ? 

19.  If  a  ship  sail  96  miles  each  day,  how  far  will  she  sail  in 
27  days  ? 

20.  If  a  ship  sail  2784  miles  in  29  days,  how  far  will  she  sail 
on  an  average  each  day  ? 

MENTAL   EXERCISES. 

1.  If  56  yards  of  calico  will  make  7  dresses,  how  many  yards 
will  make  1  dress  ? 

2.  If  9  yards  of  calico  will  make  a  dress,  how  many  yards 
•will  make  8  dresses  ? 

3.  If  9  yards  of  calico  will  make  a  dress,  how  many  dresses 
will  45  yards  make  ? 

4.  Jane  has  26  yards  of  ribbon  for  trimming  her  dress,  and 
Anna  17,  how  many  more  has  Jane  than  Anna  ? 

5.  How  many  yards  have  both  together  ? 

6.  At  $4  a  week,  how  many  weeks  can  a  person  board  for 
$416? 

7.  At  $6  a  week,  how  much  will  26  weeks'  board  cost  ? 

8.  If  23  weeks'  board  cost  $115,  what  is  the  price  per  week? 

EXEECISE  m. — FOR  THE   SLATE  OR   BLACKBOARD. 

21.  In  a  large  hotel,  857  pounds  of  beef  are  consumed  daily  ; 
how  many  pounds  will  be  consumed  in  365  days. 

22.  If  1000  men  consume  137970  pounds  of  beef  in  365  days, 
how  much  will  they  consume  in  a  day  ? 

23.  If  378  pounds  of  beef  be  consumed  daily,  how  long  will 
it  take  to  consume  137592  pounds  ? 

24.  At  $18  a  barrel,  how  many  barrels  of  sugar  can  be  bought 
for  $32166  ? 


54  PKOMISCUOUS  EXAMPLES. 

25.  At  $18  a  barrel,  how  much  will  1700  barrels  of  sugar 
cost? 

26.  If  1700  barrels  of  sugar  cost  $32300,  what  is  the  price 
per  barrel  ? 

27.  A  merchant  bought  320  barrels  of  molasses  for  $4800  ; 
2000  barrels  for  $28000  ;  1900  barrels  for  $29730  ;  how  much 
did  they  all  cost  ? 

28.  At  $15  a  barrel,  how  many  barrels  of  molasses  can  be 
bought  for  $29730  ? 

29.  At  $14  a  barrel,  how  much  will  2000  barrels  of  molasses 
cost? 

30.  If  320  barrels  of  molasses  cost  $4800,  what  is  the  price 
per  barrel  ? 

EXERCISE  rv. 

31.  At  $67  an  acre,  how  many  acres  of  land  can  be  bought 
for  $122878  ? 

32.  At  $53  an  acre,  how  much  will  234  acres  of  land  cost  ? 

33.  If  872  acres  of  land  cost  $47088,  what  is  the  price  per 
acre  ?  * 

34.  If  a  person  travel  26  miles  a  day,  how  far  will  he  travel 
in  14  days  ? 

35.  If  a  person  travel  52  miles  a  day,  how  long  will  it  take 
him  to  travel  728  miles  ? 

36.  If  a  person  travel  390  miles  in  15  days,  at  what  rate  per 
day  does  he  travel  ? 

37.  If  a  man  travel  1020  miles  the  first  week,  and  965  the 
next,  how  far  will  he  travel  in  the  two  weeks  ? 

38.  If  a  man  travel  away  from  home  3400  miles,  and  765 
miles  on  his  return,  how  far  from  home  will  he  be  ? 

39.  A  manufacturer  paid  19  journeymen  $57  apiece,  what 
was  the  amount  paid  ? 

40.  A  manufacturer  paid  his  journeymen  $1140,  and  each 
one  received  $57,  how  many  were  there  ? 

EXERCISE  v. 

41.  If  235  barrels  of  mackerel  cost  $3055,  what  is  the  price 
per  barrel  ? 


MENTAL   EXERCISES.  55 

42.  At  $14  a  ban-el,  how  much  will  235  ban-els  of  mackerel 
cost? 

43.  At  $13  a  ban-el  how  many  barrels  of  mackerel  can  be 
bought  for  $3042  ? 

44.  A  speculator  having  $15000  lost  $7000,  and  afterwards 
gained  $9653,  how  much  did  he  then  have  ? 

45.  At  $19  each,  how  much  will  346  overcoats  cost  ? 

46.  If  345  overcoats  cost  $6555,  what  is  the  price  of  each  ? 

47.  At  $18  each,  how  many  overcoats  can  be  bought  for 
$6228. 

48.  At  13  cents  a  pound,  how  many  pounds  of  cheese  can  be 
bought  for  8775  cents. 

49.  At  24  cents  a  bushel,  how  much  will  496  bushels  of  ap- 
ples cost  ? 

50.  If  a  man  travel  28  miles  a  day,  how  long  will  it  take  him 
to  travel  4256  miles  ? 

EXERCISE  VI. 

51.  A  huckster  carried  6867  melons  to  market  in  27  loads ; 
how  many  in  each  load  ? 

52.  If  a  huckster  carries  325  melons  at  a  load,  how  many 
will  he  carry  in  21  loads  ? 

53.  A  huckster  carries  337  melons  at  a  load  ;  in  how  many 
loads  will  he  carry  8175  melons  ? 

54.  A  huckster  took  to  market  at  one  time  179  cabbages ; 
at  another  268 ;   at  another  947 ;    and  at  another  144.     He 
finally  sold  1000  and  brought  back  the  rest  for  his  cattle  ;  how 
many  did  he  bring  back  ? 

55.  At  63  cents  a  basket,   how  much  will  325   baskets  of 
peaches  cost  ? 

56.  At  65  cents  a  basket,  how  many  baskets  of  peaches  can 
be  bought  for  $208  ? 

57.  If  325  baskets  of  peaches  cost  $195,  what  is  the  price 
per  basket  ? 

58.  What  cost  45  cows,  at  $40  each  ? 

59.  A  carriage-maker  sold  77  carriages,  for  $212  each ;  how 
much  did  he  receive  for  all  of  them  ? 


56  UNITED   STATES  MONEY. 

60.  A  speculator  bought  1400  acres  of  land,  at  $56  an  acre, 
and  selling  it  he  gained  $6600  ;  for  what  did  he  sell  it  per  acre  ? 

EXERCISE  VII. 

61.  John's  arithmetic  contains  296  pages,  and  he  wishes  to 
review  it  for  examination  in  18  days  ;  how  many  pages  must 
he  review  each  day  ? 

62.  After  reviewing  it  14  days,  how  many  pages  would  be 
left? 

63.  James  reviewed  18  pages  a  day  in  the  same  arithmetic  ; 
in  how  many  days  could  he  finish  it  ? 

64.  A  country  merchant  went  to  New  York  to  buy  goods, 
and  paid  for  them  in  cash  1215  dollars  ;  in  notes  1238  dollars, 
in  barter  2512  dollars  ;  all  his  expenses  were  65  dollars ;  he  sold 
them  for  6000  dollars  ;  what  did  he  gain  ? 

65.  A  horse  dealer  having  2549  dollars,  bought  21  horses,  and 
after  paying  for  them  had  113  dollars  left ;    what  was  the 
average  price  of  the  horses  ? 

66.  At  200  dollars  each,  how  many  horses  can  be  bought  for 
3000  dollars  ? 

67.  At  150  each,  how  much  will  16  horses  cost  ? 

68.  If  a  family's  expenses  are  18312  dollars  in  24  years  ;  how 
much  do  they  average  a  year  ? 


•V 


UNITED   STATES   MONEY. 


Art.  22. — United  States  or  Federal  Money  is  the  legal 
money  of  the  United  States. 
It  consists  of  Eagles,  Dollars,  Dimes,  Cents,  and  Mills. 

Its  Coins  are  in— 

Cold.—  Double  Eagle,  Eagle,  Half  Eagle,  Quarter  Eagle,  Three  Dol- 
lars, and  Dollar. 

Silver.— Dollar,  Half  Dollar,  Quarter  Dollar,  Dime,  (Ten  Cents,) 
Half  Dime,  Three  Cents. 

United  States  Money  is  a  species  of  compound  numbers  ;  but  may 
also  be  treated  much  like  simple  numbers,  since  it  increases  in  the 
same  ratio. 


UNITED   STATES  MONEY.  57 

TABLE. 

10  mills  (m.)  make  1  cent,  (ct.) 

10  cents  make  1  dime. 

10  dimes,  or  100  cents  make  1  dollar.  ($.) 
10  dollars  make  1  eagle, 

Art,  23,— Aliquot  Parts  of  a  number  are  such  as  will 
divide  it  without  a  remainder. 

ALIQUOT  PAETS   OF  U.    S.    MONEY. 


5      mills  =  %  cent. 
10     cents  =  fL   dollar. 


cents  =  %  dollar. 


50 


75 


20        "     =  i 
25        "     =  I 

Art,  24, — NOTATION  OF  U.  S.  MONEY. 
EULE. —  Write  the  dollars  as  in  simple  numbers,  with  a  point 
(.)  on  the  right ;    next  to  this,  if  there  are  cents  and  mills, 
write  two  figures  or  ciphers  for  cents,  and  then  one  figure  or 
cipher  for  mills. 

EXAMPLES  TO   BE  WRITTEN. 

1.  Ten  dollars  fifteen  cents  and  seven  mills.  $10,157 

2.  Seven  dollars  seven  cents  and  seven  mills. 

3.  Sixty  dollars  and  six  mills. 

4.  Fifty  dollars  fifty  cents  and  five  mills. 

5.  Nine  dollars  six  cents  and  eight  mills. 

6.  Sixty-three  dollars  four  cents  and  two  mills. 

7.  One  hundred  dollars  and  twenty-five  cents. 

8.  Two  hundred  and  ten  dollars  and  five  mills. 

9.  Seventy-five  dollars  two  cents  and  one  mill. 

10.  Five  hundred  dollars  and  fifty  cents. 

11.  Twelve  and  a  quarter  dollars. 

12.  Sixty-one  dollars  thirty-seven  and  a  half  cents. 

13.  Twenty  and  a  half  dollars. 

14.  Seventy -five  and  three  quarter  dollars. 

15.  One  thousand  dollars  twelve  and  a  half  cents. 


68  UNITED   STATES  MONEY. 

Art.  25* — NUMERATION  OF  U.  S.  MONEY. 

RULE. — Read  the  figures  before  the  separating  point  as  dol- 
lars ;  the  next  two  (if  there  are  any]  as  cents,  and  the  third  as 
mills. 

EXAMPLES  TO   BE   BEAD. 

1.  $12. 375.     Twelve  dollars,  thirty-seven  cents  and  five  mills 
or  half  a  cent. 

2.  $10.25  7.  $250.043  12.  $  21.25 

3.  $  9.375  8.  $125.000  13.  $202.458 

4.  $12.  9.  $87.00  14.  $405.50 

5.  $12.00  10.  $121.  15.  $700. 

6.  $12.000  11.  $63.405  16.  $700.00 

Art*  26* — REDUCTION  OF  U.  S.  MONEY. 
Reduction  of  U.  S.  Money  is  changing  dollars  to  cents, 
and   cents   to    mills,   or    mills  to   cents,  and  cents  to 
dollars,  &c. 

Since  there  are  100  cents  in  1  dollar,  and  10  mills  in  1  cent,  any 
number  of  dollars  is  equal  to  as  many  hundred  cents  or  thousand 
mills  ;  thus, 

$1.=100  cts.  =1000  m.  $12.  =1200  cts.  =12000  m.  $3.87=387 
cts.  =3870  m.  $4.375=437  cts.  5  m.=4375  m. ;  hence, 

RULES. — To  reduce  dollars  to  cents,  multiply  by  100  or 
annex  two  ciphers. 

To  reduce  cents  to  mills,  multiply  by  10  or  annex  one 
cipher. 

To  reduce  dollars  to  mills,  annex  three  ciphers. 

To  reduce  dollars  and  cents  to  cents,  or  dollars,  cents,  and 
mills  to  mills,  remove  the  separating  point. 

This  is  the  same  as  reducing  the  dollars  and  adding  the  cents  or 
mills. 

Again,  since  10  mills  make  1  cent,  and  100  cents  make  1  dol- 
lar, every  10  mills  in  any  number  make  1  cent,  and  every  thousand 
mills  or  hundred  cents  make  1  dollar  ;  thus, 

1000  m.=100  cts.=$l.  15000  m.=1500  cts.  =$15.  250  cts.= 
$2.50.  6375  m.=637  cts.  5  m.=$6.375  ;  hence, 


UNITED   STATES  MONEY.  59 

BULES. — To  reduce  mills  to  cents,  divide  by  10,  or  point  off 
the  right  hand  figure. 

To  reduce  cents  to  dollars,  divide  by  100,  or  point  off  two 
figures. 

To  reduce  mills  to  dollars,  point  off  three  figures. 

MENTAL  EXERCISES. 

How  many  mills  in  2  cents  ?  3?  5?  8?  9?  10?  13? 
16  ?  20  ?  23  ?  28  ?  31  ?  40  ?  56  ?  75  ? 

How  many  cents  in  $2?  $4?  $5?  $7?  $10?  $15? 
$20?  $24?  $36?  $42?  $50?  $75?  $87?  $90?  $100? 

How  many  cents  in  10  mills  ?  20  ?  40  ?  50  ?  65  ?  24  ? 
30?  36?  45?  50?  100?  210?  750? 

How  many  dollars  in  200  cents  ?    400?    500?    800?    1000? 

How  many  cents  in  20  dimes  ?    30  ?    50  ?    80  ? 

How  many  dimes  in  20  cents  ?     30  ?     50  ?    80  ? 

How  many  cents  in  $3  ?  20  dimes  ?  30  mills  ?  $5  ?  50 
mills  ?  50  dimes  ?  $60  ?  60  dimes  ?  60  mills  ? 

How  many  dollars,  &c.,  in  125  cents?  125  dimes?  2000 
mills?  3250  mills?  375  cents  ? 

How  many  mills  in  25  cents  and  3  mills  ?  20  cts.  5  mills  ? 
7  cts.  5  mills  ?  80  cts.  9  mills  ? 

How  many  cents  in  $2.37  ?  $6.25  ?  75  dimes  ?  75  mills  ? 
$75? 

EXAMPLES  FOE  THE   SLATE. 

Reduce  or  change — 


1.  $25  to  cents. 

2.  10250  cents  to  dollars,  &c. 

3.  250  mills  to  cents. 

4.  1100  cents  to  mills. 

5.  1100  cents  to  dollars. 

6.  $3.75  to  mills. 

7.  $10.25  to  cents. 

8.  170  cents  to  miUs. 

9.  170  mills  to  cents. 
10.  170  cents  to  dollars. 


11.  $70  to  mills. 

12.  $60  to  cents. 

13.  $24. 25  to  mills. 

14.  $30.375  to  mills.    ' 

15.  $35.50  to  cents. 

16.  75375  nulls  to  dollars. 

17.  75375  cents  to  dollars. 

18.  34000  mills  to  cents. 

19.  $45  to  mills. 

20.  $250  to  mills. 


60  UNITED   STATES  MONEY. 


30.  675  mills  to  cents. 

31.  $37  to  cents. 

32.  $16. 37  to  cents. 

33.  $21. 04  to  mills. 

34.  13405  cents  to  dollars. 

35.  759  cents  to  mills. 

36.  287  cents  to  doUars. 

37.  $300  to  mills. 

38.  1200  mills  to  dollars. 


21.  1275  mills  to  dollars,  £c, 

22.  1000  mills  to  cents. 

23.  1000  cents  to  dollars. 

24.  1375  mills  to  dollars,  &c. 

25.  1000  dimes  to  dollars. 

26.  $927.25  to  cents. 

27.  3760  cents  to  dollars. 

28.  1275  cents  to  mills. 

29.  1325  mills  to  dollars. 


Art,  27.— Application  of  the  Fundamental  Rules  to 
U,  S,  Money, 

Since  numbers  in  U.  S.  money  increase  from  right  to 
left  in  a  ten-fold  ratio,  the  same  as  simple  numbers,  they 
may  be  added,  subtracted,  multiplied,  and  divided  by 
nearly  the  same  rules. 

Art,  28, — ADDITION  OF  U.  S.  MONEY. 

EULE. —  Write  the  numbers  so  that  the  separating  points  will 
be  under  one  another,  and  proceed  as  in  simple  Addition. 

EXAMPLE  1.— Add  $5.125,  $17.062,  $10.43,          Process. 
$8.055,  $15.706.  \^ 

2.  Add  $18.15,  $24.45,  $7.21,  $9.38,  $11.33.  17.06  2 

3.  Add  $19.041,  $17.315,  $112.18,  $75.873,  10.43 
$60.50.  ^l  I 

4.  Add  $44.76,  $28.19,  $18.657,  $270.508,        An8m  ^^37~8 
$87.60;  $67.005. 

5.  Add  $10.625,  $112.35,  $1.75,  $11.875,  $100,  $17.37. 

6.  What  is  the  sum  of  $21  lOcts.,  $17  4cts.  6m.,  $23  17cts. 
3m.,  $19  18cts.  6m.,  $25,  $16  8  cts.,  $15  5m.? 

7.  Add  $200,  $43.875,  $56.937,  $18.50.  $12.315. 

8.  What  is  the  amount  of  $304  50cts.,  $304  4m.,  $820  35cts. 


ADDITION  OF  U.    S.    MONEY.  61 

9.  What  is  the  sum  of  $25  Sets.,  $40  21ots.  3m.,  $108  5m., 
$63  4cts.,  $312  let.  7m.,  $1000  15cts.,  $50  8m.? 

10.  Add  $6  Gets.  3m.  t  $14  17cts.,  $21  Sets.  6m.,  $25  50cts., 
$17  8m.,  $100  lOcts.  3m.,  $1  let.  1m.,  $10  lOcts.  7m. 

11.  Add  $5  4cts.  3m.,  $1  14cts.,  $98,  $2  2m.,  Sots.  3m.,  $15 
16cts.  4m. 

12.  What  is  the  amount  of  $300,  $4  4cts,  $50  5m.,  $70  Tots., 
$45  5m.  ? 

13.  What  is  the  sum  of  35  dollars  6  cents  7  mills,  $11  4cts. 
6m.,  $17  18cts.  9m.,  $400  83cts.,  $12  20cts.  2m.? 

14.  Add  3  dollars  12  cents  5  mills,  $50  50cts.,  $300  6m., 
75cts.,  $75  Tots.  5m.,  $201  3cts. 

15.  What  is  the  sum  of  $18%,  $12}£,  $6^,  $%,  $5^?    (See 
Table  of  Aliquot  Parts.) 

16.  What  is  the  amount  of  9  dollars  62}£  cents,  87)£cts, 
$15^,  $108  62)^  cts.,  $1  Sets,  $27%,  $63  12>^cts.? 

17.  Add  10}£  dollars,  87>£  cents,  $105  62^cts.,  $16^  37)£ 
cents,  $21%. 

18.  Add  9  dollars  12}£  cents,  $6  3m.,  $28  87>£cts,  $56  5cts. 
5m. 

19.  What  is  the  sum  of  $39,  $109  12}£cts.,  5m,  $5  5m.,  $1 


20.  What  is  the  amount  of  $67  12)£  cts.,  $60%,  $62^,  $ 


Add  the  following  numbers  in  U.  S.  money  — 

21.  Three  hundred  dollars  and  three  cents, 
Three  dollars  and  three  mills, 

Five  hundred  dollars, 
Five  hundred  cents, 
Five  hundred  mills. 

22.  Eighty-five  doUars, 

Sixty  dollars  sixty-two  and  a  half  cents, 
Thirty-seven  and  a  half  cents, 
Forty  dollars  four  cents  and  five  mills, 
Forty  cents  and  four  mills, 
Forty-four  mills. 


62  UNITED   STATES  MONEY. 

23.  Seventy  dollars, 

Five  dollars,  eighty-seven  and  a  half  cents, 
Fifty  dollars  fifty  cents  and  five  mills, 
Six  and  three-quarter  dollars, 
Five  and  a  half  dollars  and  a  half  cent. 
Ten  and  one  quarter  dollars. 

24.  Two  dollars  and  two  mills, 

Seven  dollars  eighty-seven  and  a  half  cents, 
Nine  dollars  thirteen  cents  and  three  mills, 
Sixty-seven  dollars  and  eight  mills, 
Four  dollars  and  seventy-five  cents, 
One  and  three-quarter  dollars. 

25.  Seven  dollars  and  eighty  cents, 
Twelve  dollars  and  twenty-five  cents, 
Ten  dollars  and  two  mills, 
Sixty-five  dollars, 

Six  cents  and  five  mills, 

One  dollar  one  cent  and  one  mill. 

26.  Ten  eagles  ten  dollars  ten  dimes  ten  cents  and  ten 

27.  One  half  eagle  one  half  dollar  and  one  half  cent. 

28.  Thirty-seven  and  a  half  dollars, 
Thirty-seven  and  a  half  cents, 
Twenty-four  and  three-quarter  dollars, 
Six  and  a  quarter  dollars  and  a  half  cent, 
Twelve  and  a  half  cents. 

29.  A  family  has  paid  for  beef  $19.15,  flour  $17.375,  butter 
$10.125,  and  sugar,  $4.65,  what  is  the  amount  ? 

30.  A  farmer  bought  a  horse  for  $135,  a  pair  of  oxen  for 
$97.375,  a  cow  for  $35,  and  20  sheep  for  $50,  how  much  did  he 
pay  for  them  all  ? 

31.  A  young  man  bought  a  suit  of  clothes  for  $56,  a  watch 
for  $87>£,  a  watch  chain  for  $12%,  and  a  pair  of  gloves  for 

cents,  what  did  they  all  cost  ? 

32.  A  young  lady  bought  a  silk  dress  for  $26,  a  shawl  for 
$18%,  a  bonnet  for  $7^,  and  a  pair  of  gaiters  for  $3.37>o> 
what  lid  they  all  cost  ? 


SUBTRACTION  OF  U.    S.    MONEY.  63 


Art.  29. — SUBTEACTION  OF  U.  S.  MONEY. 

EULE. —  Write  the  numbers  so  thai  the  separating  points  mill 
be  under  each  other,  and  proceed  as  in  simple  Subtraction. 

EXAMPLE  1.  From  $125  take  $37.053.  Proem. 

2.  From  $39.25  take  $16.246.  125.  wTo 

3.  From  $127.384  take  $15.60.  37J05  3 

4.  From  $95.28  take  $45.183.  Ans.  $87.94  7 

5.  From  $118.05  take  $67.45. 

6.  From  $95  take  $33. 60. 

7.  From  $25  take  25  cents. 

8.  From  $100  take  100  cents. 

9.  From  $10  take  10  cents. 

10.  From  $1  take  1  cent. 

11.  From  1  cent  take  1  mill. 

12.  From  $5  take  5  mills. 

13.  A  man  paid  $175  for  a  carriage,  and  $162}^  for  a  horse, 
how  much  more  did  he  pay  for  the  carriage  than  the  horse  ? 

14.  A  merchant  bought  a  hogshead  of  molasses  for  $26% 
and  sold  it  for  $35,  how  much  did  he  gain  ? 

15.  A  young  man  sold  his  watch  for  $37^,  and  it  cost  $45. 
how  much  did  he  lose  ? 

16.  A  lady  having  $50,  spent  $27.62}£  in  shopping,  how 
much  was  left  ? 

17.  A  laborer  has  earned  $100,  and  been  paid  $53.87^,  how 
much  is  still  due  to  him  ? 

18.  A  man  owed  $500,  and  has  paid  $263.62)^,  how  much 
does  he  still  owe  ? 

19.  A  person  having  bought  a  bill  of  goods  amounting  to 
$7.12^,  gave  in  payment  a  ten  dollar  bill,  how  much  change 
did  he  receive  ? 

20.  One  man  earns  $1.62)^,  and  another  $1%  a  day,  how 
much  more  does  one  earn  than  the  other  ? 


64  UNITED   STATES  MONEY. 


Art,  30,— Multiplication  of  U,  S,  Money, 

EULE. — Proceed  as  in  simple  Multiplication,  and  place  the 
separating  point  as  far  from  the  right  as  it  is  in  the  multipli- 
cand, sometimes  used  as  the  multiplier. 

Process. 

$8.625 

15 

EXAMPLE  1.— Multiply  $8.625  by  15.  13125 

8625 
Ans.  $129.375 


Multiply  — 
(2.)  $112.08        by     7 
(4.)    $94.375        "      9 
(6.)     $65.             "    12 
(8.)  $100.             "  100 
(10.)           75cts.    "    14 
(12.)            12>£cts"    10 
(14.)  $8^               "      5 

(3.)     $1.25        by     10 
(5.)  $12.50         "    100 
(7.)  $48.375        "      35 
(9.)  $10.             "    100 
(11.)             5  mis"      25 
(13.)  $12^           "      20 
(15.)  $18%           "  1000 

In  multiplying  U.  S.  money  by  10,  100,  &c. ,  it  is  sufficient  to  re- 
move the  separating  point  as  many  places  to  the  right  as  there  are 
ciphers  in  the  multiplier  ;  as  $4.50><100=  $450. 

16.  At  $1.25  a  bushel,  how  much  will  20  bushels  of  corn  cost  ? 

17.  At  62)^  cents  a  bushel,  what  will  15  bushels  of  apples 
cost? 

18.  If  an  acre  of  land  cost  $87}^,  how  much  will  100  acres 
cost? 

19.  If  a  cord  of  wood  cost  $6%,  how  much  will  12  cords  cost  ? 

20.  At  $5.62)^  a  yard,  how  much  will  3  yards  of  cloth  cost  ? 

21.  At  12>£  cents  a  quart,  how  much  will  7  qts.  of  cherries 
cost? 

22.  If  one  doz.  eggs  are  worth  $^,  how  much  are  100  doz. 
worth  ? 

23.  If  a  pound  of  butter  is  worth  22>£  cts.,  what  are  14 
pounds  worth  ? 

24.  At  $10%  a  ton,  what  are  10  tons  of  hay  worth  ? 


DIVISION  OF   U.    S.    MONEY.  65 


Art,  31,— Division  of  U,  S.  Money. 

RULE. — Proceed  as  in  simple  Division,  observing  that  either 
the  divisor  or  quotient  must  be  of  the  sam,e  name  as  the  divi- 
dend, reduced  if  necessary  to  cents  or  mills,  and  have  the  sep- 
arating point  in  the  same  place  ;  while  the  other  has  no  sepa- 
rating point  except  in  decimal  fractions. 

The  dividend  is  the  price  of  the  whole  quantity. 

The  price  of  the  whole  divided  by  the  quantity  gives 
the  price  of  each  part  ;  or 

The  price  of  the  whole  divided  by  the  price  of  each 
part  gives  the  quantity. 

$5.00-^100  Ibs.,  etc.=5  cts.  $5.00-f-.05  cts.=100 
Ibs.,  etc. 

EXAMPLE  1.— Divide  $316.753  by  5.  5)$316.753 

Process.— Short  Division—       Ans.  $63.350+3 

$    c.    $  c.m. 

Ex.  2.— Divide  $225.50  by  18.  18)225.50(12.527+ 

Process. — Long  Division — 

4:0 

If  there  is  a  remainder  after  dividing  any  given        36 
number  of  dollars  or  cents,  reduce  the  dollars  to        "95" 
cents  and  the  cents  to  mills. 

50~~ 
36 

140  mills. 
126 
U 

c.m.  $    c.'mAns. 

E*.3.-Dmde  too  by  62^  cents  =  625     625H 
mills.  2250 

Process. —  1875 

~3750 
3750 


66  UNITED   STATES   MONEY. 


Divide — 

Ex.4.  $124.64  by  $3.28,  or  into  38  equal  parts. 

5.  $62.64  "  $7.83,  "  8            " 

6.  $108.837  "  $12.093,  "  9 

7.  $1862.42  "  $35.14,  "  53 

8.  $368.288  "  $23.018,  "  16 

9.  $2.125  "  $0.125,  "  17 

10.  $2.50  <•  $0.25,  "  10            " 

To  divide  U.  S.  Money  by  10,  100,  Ac.,  it  is  sufficient  to  remove 
the  separating  point  as  many  places  to  the  left  as  there  are  ciphers 
in  the  divisor  ;  thus,  $45-^-10G=$0.45. 

Ex.  11.  At  12^  cents  a  pound,  how  many  pounds  of  sugar 
can  be  bought  for  $2.00. 

12.  If  24  pounds  of  sugar  cost  $3.00,  what  is  the  price  per 
pound  ? 

13.  At  28  cents  a  pound,  how  many  pounds  of  butter  can 
be  bought  for  $2.24? 

14.  If  9  pounds  of  butter  cost  $2.52,  what  is  the  price  per 
pound? 

15.  At  65  cents  a  bushel,  how  many  bushels  of  corn  can  be 
bought  for  $130'? 

16.  If  200  bushels  of  corn  cost  $120,  what  is  the  price  per 
bushel  ? 

17.  At  37£  cents  a  bushel,  how  many  bushels  of  oats  can  be 
bought  for  $21  ? 

18.  If  60  bushels  of  oats  cost  $24,  what  is  the  price  per 
bushel  ? 

19.  If  a  bushel  of  wheat  cost  $1.37^,  how  many  bushels  can 
be  bought  for  $49.50  ? 

20.  If  40  bushels  of  wheat  cost  $55,  what  is  the  price  of 
one  bushel  ? 

21.  At  $li^  a  day,   in  how  many  days  can  a  man  earn 
$25? 

22.  If  a  man  earn  $20  in  16  days,  how  much  IB  it  a  day  ? 


ALIQUOT   PARTS   OF  A  DOLLAR  67 

Art.  32.— Aliquot  Parts  of  a  Dollar. 

(See  Table  of  U.  8.  Money.} 

When  the  price  of  anything  is  an  aliquot  part  of  $1, 
it  shortens  the  operation, 

To  divide  by  the  number  of  parts,  instead  of  multiplying 
by  the  number  of  cents,  and  to  multiply  instead  of  dividing. 

EXAMPLE  1.  What  will  49  yards  of  calico  cost,  at  25  cts.  a 
yard  ? 

Process.— Since  1  yard  costs  25  cts.—  ($£,)  1  quar-    4)49 

ter  of  a  dollar,  49  yards  will  cost  1  quarter  of  49 

dollars.  $12i=12.25 

Ex.  2.  How  many  yards  of  calico  can  be  bought  for  $10,  at 
25  cts.  a  yard  ? 

Process. — Since  25  cents,  or  ($£)  will  buy  one  10 

yard,  $1  will  buy  4  yards,  and  $10  will  bay  (4  times  4 

10)  40  yards.  4$ 

EXAMPLES. 

3.  At  20  cts.  each,  -what  will  400  writing  books  cost  ? 

4.  At  25  cents  each,  how  many  writing  books  can  be  bought 
for  $5  ? 

5.  At  33^  cts.  a  gallon,  what  cost  30  gallons  of  vinegar  ? 

6.  At  33^  cts.  a  gallon,  how  many  gallons  of  vinegar  can  be 
bought  for  $12  ? 

7.  At  50  cents  a  bushel,  how  many  bushels  of  apples  can  be 
bought  for  $25  ? 

8.  At  50  cts.  a  bushel,  how  much  will  100  bushels  of  apples 
cost? 

9.  At  12^  cts.  a  pound,  how  many  pounds  of  rice  can  be 
bought  for  $6  ? 

10.  At  12-|  cents  a  pound,  how  much  will  16  pounds  of  rice 
cost? 

11.  If  a  boy  earn  33}^  cts.  a  day,  how  much  will  he  earn  in 
6  days  ? 

12.  If  a  boy  earn  33^  cts.  a  day,  in  how  many  days  will  he 
earn  $1  ? 


68  UNITED   STATES   MONEY. 

Art,  33,— Price  per  Hundred  or  Thousand, 

When  the  given  price  is  per  hundred,  call  the  dollars 
cents  ;  when  per  thousand,  call  them  mills,  which  will  be 
the  price  of  one  of  the  things  specified. 

EXAMPLE  1. — What  will  a  bale  of  hay  weighing  156  pounds 
cost  at  $1  per  hundred  ? 

Process.— $1=100  cts.,  and  100  cts.  per  hundred 
pounds  is  1  cent  per  pound,  and  156X-01— $1.56. 

Ans.  $1.56 

Ex.  2.  What  cost  12500  shingles  at  $18  per  thousand  ? 

3.  What  cost  7500  bricks  at  $9  per  thousand  ? 

4.  At  $9  per  thousand,  how  many  bricks  can  be  bought  for 
$63? 

5.  What  cost  56  pounds  of  flour  at  $4  per  hundred  ? 

6.  What  cost  615  feet  of  pine  boards,  at  $21  per  thousand  ? 

Examples  of  this  kind,  in  which  the  price  is  cents,  in- 
volve decimal  fractions,  but  the  process  is  the  same. 


Art,  34,— Promiscuous  Examples  in  the  Fundamental 
Rules,  including  U.  S,  Money. 

EXERCISE  I. 

1.  A  young  man  bought  a  horse  for  $150 ;    a  watch  for 
$53.875  ;  a  suit  of  clothes  for  $46.937  ;  a  hat  for  $4.50 ;  a  pair 
of  boots  for  $4.00,  and  some  other  things  for  $2.313  ;  what  was 
the  amount  ? 

2.  From  $100  subtract  $1,  1  cent  and  1  mill. 

3.  At  25  cents  a  yard,  how  many  yards  of   ribbon  can  be 
bought  for  $4. 

4.  At  25  cts.  a  yard,  how  much  will  12  yards  of  ribbon  cost  ? 

5.  If  20  yards  of  ribbon  cost  $5,  what  is  the  price  per 
yard? 

6.  At  $8.05  a  ton,  how  much  will  20  tons  of  hay  cost  ? 

7.  At  34  cents  a  yard,  how  many  yards  of  muslin  can  be 
bought  for  $30.26  ? 


UNITED   STATES   MONEY.  6£ 

8.  In  a  case  of  broadcloth  there  are  19  pieces,  containing  in 
all  437  yards  ;  how  many  yards  in  a  piece  on  an  average  ? 

9.  A  farmer  carried  to  market  20  loads  of  oats,  and  each, 
load  contained  75  bushels  ;  how  many  bushels  in  all  ? 

10.  A  farmer  had  1200  bushels  of  wheat,  and  could  carry 
50  bushels  at  a  load  ;  how  many  loads  were  there  ? 

11.  A  lady  went  shopping  with  $5  in  her  purse  ;   she  paid 
75  cents  for  a  collar ;  $1,50  for  kid  gloves;  50  cts.  for  ribbon 
and  25  cts.  for  needles  and  pins  ;  how  much  had  she  left  ? 

12.  At  Si.  60  a  day,  how  much  will  a  man  earn  in  40  days  ? 

13.  At  $1. 121  a  bushel,  how  many  bushels  of  wheat  can  be 
bought  for  $208. 

14.  At  431  cts.  a  bushel,  what  will  750  bushels  of  buckwheat 
cost? 

15.  A  farmer  owed  a  merchant  $500,  and  paid  him  435  bush- 
els of  oats  at  45  cts.  a  bushel ;  how  much  does  he  still  owe  ? 

EXERCISE  n. 

16.  At  7^  cts.  a  quart,  how  many  quarts  of  cherries  can  be 
bought  for  $1.35  ? 

17.  If  a  clerk's  salary  is  $800  a  year,  how  much  is  it  for 
each  day  he  is  employed  in  business  (313  in  the  year)  ? 

18.  At  $2.25  a  day,  how  much  will  a  laborer  earn  in  313 
days  working  time  in  a  year  ? 

19.  At  $17.565  an  acre,  how  many  acres  of   land  can  b© 
bought  for  $2722. 575? 

20.  If  3  men  gain  $1000,  what  is  each  one's  equal  share  ? 

21.  A  lady  having  $26,  bought  a  silk  dress  for  $13.10  ;   a 
shawl  for  $6,  and  gloves  for  75  cts  ;  how  much  had  she  left  ? 

22.  What  cost  8  pieces  of  calico,  each  containing  19  yards, 
at  23  cts.  a  yard  ? 

23.  At  $5.67  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $136.08  ? 

24.  If  168  lambs  cost  $451.92,  what  is  the  price  of  each  ? 

25.  What  cost  17  firkins  of   butter,   each   containing  51 
pounds,  at  14  cents,  7  mills  per  pound  ? 


70  FUNDAMENTAL   RULES. 

EXERCISE  TTT. 

26.  At  $12  a  barrel,  how  many  barrels  of  flour  can  be  bought 
for  $1512  ? 

27.  If  670  pounds  of  cheese  cost  $87.10,  what  is  the  price 
per  pound  ? 

28.  What  number  multiplied  by  9  will  produce  315  ? 

29.  At  $8  a  ton,  how  many  tons  of  coal  can  be  bought  for 
$1728  ? 

30.  A  farmer  sold  his  pork  for  $21.75,  and  received  sugar, 
$3.75;  molasses,  $2.50;  tea,   $1.35;    cheese,  $1;    pepper,  25 
cts. ;   ginger,  18  cts. ;   the  rest  in  cash  ;  how  much  cash  did  he 
receive  ? 

31.  How  much  coffee  at  13  cents  a  pound  can  be  bought  for 
$18.59  ? 

32.  At  $1.43  a  day,  how  much  •will  a  man  earn  in  312  days  ? 

33.  If  54  bushels  of  wheat  cost  $67.50,  what  is  the  price  per 
bushel  ? 

34.  At  $10  and  5  mills  an  acre,  what  will  150  acres  of  land 
cost? 

35.  If  17  bags  of  coffee,  each  weighing  51  pounds,  cost 
$127.449,  what  is  the  price  per  pound  ? 

EXERCISE  IV. 

36.  If  137  shares  of  bank  stock  are  worth  $17125,  what  is  a 
share  worth  ? 

37.  If  1000  men  consume  856  pounds  of  beef  in  a  day,  how 
many  pounds  will  last  them  365  days  ? 

38.  At  $34  a  barrel,  how  many  barrels  of  sugar  can  be  bought 
for  $128.690  ? 

39.  A  lady  bought  a  cloak  for  $25.125.  a  muff  for  $12.375,  a 
bonnet  for  $9.15,  and  gave  the  merchant  a  $50  bill ;  how  much 
change  was  due  her  ? 

40.  What  cost  24  arithmetics,  at  $0.37£  each  ? 

41.  At  $0.37^  each,  how  many  arithmetics  can  be  bought  for 
$13.50  ? 

42.  If  18  arithmetics  cost  $6.75,  what  is  the  price  of  each  ? 

43.  If  a  horse  travel  34  miles  a  day,  how  far  will  he  travel 
in  75  days  ? 


UNITED  STATES  MONEY.  71 

44.  If  a  horse  travel  34  miles  a  day,  how  long  will  it  take 
Trim  to  travel  1700  miles  ? 

45.  What  cost  8  barrels  of  sugar  containing  225  Ibs.  each,  at 
6|  cents  a  pound  ? 

EXEECISE  V. 

46.  A  drover  bought  397  cattle  for  $14689,   what  is  the 
average  price  of  each  ? 

47.  In  a  pile  there  are  237  boards,  each  containing  23  square 
feet,  how  many  square  feet  in  all  ? 

48.  A  manufacturer  has  a  contract  for  273249  yards  of  calico 
to  be  made  in  313  days ;   how  many  yards  must  he  average 
daily  ? 

49.  A  man  bought  a  lot  for  $375,  paid  for  building  a  house 
on  it  $750,  and  for  improvements  $160.87^  ;  he  then  sold  the 
place  for  $1500  ;  what  did  he  gain  ? 

50.  A  laborer  was  paid  $23. 75  for  19  days'  work  ;  how  much 
did  he  have  a  day  ? 

51.  At  $1.12^  a  day,  how  much  will  a  man  earn  in  16  days  ? 

52.  If  25  men  earn  $35.50  in  a  day,  how  much  will  they 
earn  in  50  days  ? 

53.  If  25  men  earn  $1775  in  50  days,  how  much  will  one 
man  earn  in  the  same  time  ? 

54.  At  37^  cents  a  bushel,  how  much  corn  can  be  bought 
for  $58.50  ?  " 

55.  Bought  20  pieces  of  muslin,  each  measuring  19  yards, 
for  $87.40  ;  what  was  the  price  per  yard  ? 


56.  A  farmer  paid  his  hired  man  $165  for  12  months,  how 
much  was  it  a  month  ? 

57.  A  laborer  dug  29  bushels  of  potatoes  a  day,  how  many 
did  he  dig  in  42  days  ? 

58.  A  laborer  agreed  to  dig  1212  bushels  of  potatoes  in  42 
days,  how  many  must  he  dig  a  day  on  an  average  ? 

59.  Eeduce  37500  mills  to  dollars. 

60.  A  merchant  bought  some  cloth  for  26  dollars  and  3  cents, 
a  bale  of  sheeting  for  50  dollars  and  90  cts».,  20  pieces  of  calico 
for  49  dollars,   1  cent,   12  pieces  of  merino  for  $108.14,  10 


72  FUNDAMENTAL   KULES. 

pieces  of  silk  for  $77.25,  15  pieces  of  linen  for  $83.68,  and  3 
dozen  kid  gloves  for  $40  and  8.  cents ;  what  was  the  amount  of 
the  bill  ? 

61.  A  young  man  having  $40,  paid  $20.20  for  a  coat ;  how 
much  had  he  left  ? 

62.  At  $8.245  a  yard,  how  much  will  10  yards  of  cloth  cost  ? 

63.  At  $6.25  a  barrel,  how  many  barrels  of   eggs  can  be 
bought  for  $256.25  ? 

64.  If  40  barrels  of  eggs  cost  $250,  what  is  the  price  per 
barrel  ? 

65.  Paid  16  men  $516  for  work  at  75  cts.   a  day  ;  how  many 
days  did  they  work  ? 

EXERCISE  vn. 

66.  A  steamship,  after  consuming  17,500  pounds  of  coal,  has 
30,000  pounds  left ;  how  many  had  she  at  first  ? 

67.  A  steamship  had  50000  pounds  of  coal,  and  has  con- 
sumed 27035  pounds  ;  how  "much  remains  ? 

68.  If  a  merchant  gain  $65  a  day,  how  much  will  he  gain  in 
(313  business  days)  a  year  ? 

69.  If  a  merchant  gain  18780  dollars  a  year,  how  much  will 
be  the  average  gain  a  day  for  313  business  days  ? 

70.  If  a  merchant  gain  $50  a  day,  how  long  will  it  take  him 
to  gain  $10000  ? 

71.  If  a  ship  sail  75  miles  a  day,  how  far  will  she  sail  in  60 
days? 

72.  If  a  ship  sail  4500  miles  in  50  days,  how  many  miles  will 
she  average  in  a  day  ? 

73.  If  a  ship  sail  60  miles  a  day,  how  long  will  it  take  her  to 
sail  4500  miles  ? 

74.  If  a  150  quarts  of  cherries  cost  $9375,  what  is  the  price 
per  quart  ? 

75.  At  $3.95  a  bushel,  how  many  bushels  of  timothy  seed 
can  be  bought  for  $146.15  ? 

EXERCISE  VUL 

76.  In  a  case  of  broadcloths  there  are  12  pieces,  and  in  each 
piece  49  yards  ;  how  many  yards  in  all  ? 

77.  In  another  case  there  are  12  pieces  containing  564  yards, 
how  many  yards  on  an  average  in  each  piece  ? 


UNITED   STATES   MONEY.  73 

78.  In  another  case  each  piece  contains  45  yards,  and  there 
are  in  all  540  yards,  how  many  pieces  ? 

79.  A  merchant  has  in  cash  $576.32,  notes  $135.375,  flour 
$97.10,  butter  $57.19,  for  which  he  wishes  to  purchase  goods 
amounting  to  $1000,  the  balance  to  remain  on  credit,  how 
much  will  the  balance  be  ? 

80.  If  360  laborers  receive  $405  a  day,  what  will  each  one 
of  them  receive  ? 

81.  At  $7.625  a  bushel,  how  many  bushels  of  flaxseed  can 
be  bought  for  $1807.125  ? 

82.  At  $6.625  a  bushel,  how  much  will  237  bushels  of  flax- 
seed  cost  ? 

83.  If   250  bushels  of  flaxseed  cost  $1312.50,  what  is  the 
price  per  bushel  ? 

84.  At  $5. 72  a  cord,  what  will  23  cords  of  wood  cost  ? 

85.  At  $5.75  a  cord,  how  many  cords  of  wood  can  be  bought 
for  $138. 

EXERCISE  IX. 

86.  A  farmer   sold  56  loads  of  hay,  each  weighing   1400 
pounds  ;  how  many  pounds  did  they  all  weigh  ? 

87.  A  farmer  received  $700  for  56  loads  of  hay;  what  was 
the  price  of  a  load  ? 

88.  Bought  18  barrels  of  sugar,  each  containing  235  pounds  ; 
how  many  pounds  in  all  ? 

89.  In  20  barrels  of  sugar  there  are  4115  pounds  ;  how  many 
pounds  will  they  average  ? 

90.  If  it  would  take  1  man  567  days  to  build  a  house,  in  how 
many  days  could  28  men  build  it  ? 

91.  A  merchant  bought  dry  goods  amounting  to  $5862.97, 
and  groceries  amounting  to  $1279.50;  he  paid  in  cash  $4000  and 
gave  notes  for  the  balance  ;  what  was  the  amount  of  the  notes  ? 

92.  Bought  75  books  at  $1.25 ;  how  much  did  they  all  cost  ? 

93.  Bought  25  pairs  of  shoes  for  $23.75 ;  what  was  the  price 
of  a  pair  ? 

94.  Paid  $51.75  for  9  hats;  what  was  the  average  price  of 
each  ? 

95.  What  cost  23  cases  of  boots,  at  $37.52  a  case  ? 

4 


74  UNITED   STATES  MONEY. 

96.  At  $0.375  each,  how  many  books  can  be  bought  for  $4.50  ? 

97.  What  cost  128  barrels  of  sugar,  at  $18.96  a  barrel  ? 

98.  If  254  barrels  of  sugar  cost  $2407.92,  what  is  the  price 
of  a  barrel  ? 

99.  At  $63.75  an  acre,  how  much  will  200  acres  of  land  cost  ? 

100.  If  100  acres  of  land  cost  $6375,  what  is  the  price  per  acre  ? 

Art,  35,— Bills  in  U,  S.  Money, 

A  bill  is  a  written  account  of  what  is  to  be  paid  for, 
as  goods,  labor,  &c. 

EXAMPLE  I. 

NEW  YOKE,  June  1st. 
J.  KING,  Dr. 

To  W.  BKOWN. 
51bs.  of  Tea,  at  62)£  cts $3.125 

8  "      Coffee,         at  15  ots 1.20 

3       "      Starch,         at  12}£  cts 375 

14      "      Sugar.          at  11  cts 1.54 

6  gals.  Molasses,         37^  cts 2.25 

What  is  the  amount  ?  Ans.  $8.49 

EXAMPLE  n. 

6  yds.   of  Cloth,      at  $4.37>£ $ 

18        "       Calico,     at      .21     

10  .     "       Muslin,   at      .19     

3  spools  Cotton,       at      .09     

5  sheets  Wadding,  at      .12>£ 

What  is  the  amount  ? 

EXAMPLE  m. 
10  Ibs.  of  Sugar,     at  $0.16     

5       "      Tea,          at    1.12)£ 

17      "      Butter,    at      .22^ 

9  "      Coffee,     at      .14     

2bbls.  Flour,         at    9.50     

What  is  the  amount  ? 


BILLS  IN  UNITED   STATES  MONET.  75 


EXAMPLE  IV. 


9yds.  Silk,  at  Si.  25 

15     "    Calico,  at 

20     "    Muslin,  at      .21 

7     "    Gingham,  at 

6  skeins  Silk,  at      .05 

What  is  the  amount  ? 


EXAMPLE  v. 

175  bushels  of  Wheat,         at 
300          "         Corn,  at      .81 

625          "         Oats,  at 

92          «         Buckwheat,  at      .56     , 
112  Eye,  at      .75     . 

What  is  the  amount  ? 

EXAMPLE   VI. 


1250  bushels  of  Potatoes,  at 

625          "         Turnips,    at 

172         "         Carrots,    at    .35 

85         "         Beets,       at    .68 

126  barrels  of  Apples,  at 

What  is  the  amount  ? 


EXAMPLE  vn. 

8  yds.  Merino,  at  $1.37>£  .  .  . 

13  "    Mns.  de  Laine,  at      .4A     ... 
11     "    Alpaca,  at      .75  .  ... 

1     "    Figured  Satin,  at    3.00     ... 

9  "    Col'd  Cambric,  at      .12^... 

14  "    Drab  Fringe,      "      .62)^..., 

What  is  the  amount  ? 


76  COMPOUND  NUMBERS. 


EXAMPLE   VUL 


«/ 


at 


tea,  at 


at 

5      n&n  naeice,    at 

at         .38 


6  fiaiA 


What  is  the  amount  ? 


In  like  manner  write  out  and  find  the  amounts  of  the  following 
bills: 

9.  T.   White  bought  of    L.  Camp,    3   yards  of    cloth  at 
$6.50 ;  2  yds.  of  cassimere,  at  $2.75 ;  5  yds.  cambric,  at  37)£cts. ; 

2  doz.  buttons,  at  12>£  cts. ;  3  skeins  of  silk,  at  6^  cts. 

10.  W.   Savage   bought  of  L.  Stearns,   2  bbls.  of   flour, 
at  $9.50 ;  25  Ibs.  of  sugar,  at  18%  cts.;  10  Ibs.  coffee,  at  37}£ 
cts: ;  3  Ibs.  tea,  at  $1. 12)^  ;  1  bar  of  soap,  at  15  cts. 

11.  Mrs.   Nelson  bought  of  A.  Halsted,  12  yards  of  silk, 
at  $2^' ;  3  yds.  satin,  at  $12)£  ;  6  yds.  of  cambric,  at  16  cts. ; 

3  pairs  of  hose,  at  56)^  cts. ;  8  skeins  of  silk,  at  4  cts. ;  1  doz. 
spools  of  cotton,  at  62)^  cts. 

12.  S.  Hoes  bought  of  I.  Ball  &  Co.,  4  yards  of  cloth, 
at  $6  ;  12  yds.  of  satinet,  at  87)£  cts. ;  3  woolen  shirts,  a 

6  collars,  at  20  cts.;  1  pair  gloves,  at  $1.50. 


COMPOUND  NUMBERS.  77 

COMPOUND   NUMBERS. 

Art.  36. — Compound  Numbers  are  numbers  having  dif- 
ferent denominations,  or  names,  under  a  general  name, 
to  express  one  quantity  ;  as,  pounds,  shillings,  pence  and 
farthings  under  English  or  Sterling  Money,  and  pounds, 
ounces,  etc.,  under  Weights. 

The  general  names  include  Money  "Weights  and  Meas- 
ures of  different  kinds  ;  the  different  denominations  of 
which  are  exhibited  in  the  following  tables. 

Money. 

United  States  Money  is  a  species  of  compound  num- 
bers, but,  being  much  like  simple  numbers,  has  been  al- 
ready introduced. 

Art.  37.— English  or  Sterling  Money  is  the  money  used 
in  England.  It  was  formerly  used  in  this  country,  and  is 
still  used  to  some  extent,  though  its  value  has  been 
changed,  and  varies  in  different  States. 

TABLE. 

4  farthings  (far.)  make 1  penny.  (d.) 

12  pence  1  shilling.  (s.) 


20  shillings  ..1  (£. 

(    sovereign. v 

21  shillings  1  guinea. 

£1  is  valued  at 


Weights. 

Art.  38,— Troy  Weight  is  used  in  weighing  gold,  silver, 
and  gems. 

TABLE. 

24  grains    (gr. )     make 1  pennyweight,  (pwt. ) 

20  pennyweights  1  ounce.  (oz.) 

12  ounces  1  pound.  (Ib.) 


78  COMPOUND  NUMBEBS. 

Art,  39.— Avoirdupois  Weight  is  used  in  weighing 
heavy  and  common  articles. 

TABLE. 

16  drams    (df.)    make 1  ounce.  (oz.) 

16  ounces                        1  pound.  (Ib.) 

25  pounds                        1  quarter.  (qr. ) 

4  quarters,  or  100  Ibs 1  hundred-weight,  (cwt.) 

20  hundred-weight         1  ton.  (T.) 

Formerly  28  Ibs.  made  1  qr.,  and  112  Ibs.  made  1  cwt.  1  Ib. 
(av.)=7000  grains  (troy.) 

Art,  40.— Apothecaries'  Weight  is  used  in  weighing 
medicines  at  retail.  The  pound  and  ounce  are  the  same 
as  in  Troy  Weight. 

TABLE. 

20  grains  (gr.)  make 1  scruple,  (sc.  or  B) 

3  scruples  1  dram.  (dr.  or  5) 

8  drams  1  ounce.  (oz.  or  §) 

12  ounces  1  pound.  (ft>.) 

Art.  41. — MISCELLANEOUS  TABLE  OF  WEIGHTS. 

196  Ibs.  make 1  barrel  of  flour. 

200  1  barrel  of  beef,  pork  or  fish. 

56  1  firkin  of  butter. 

60  1  bushel  of  wheat. 

56  '.  .1  bushel  of  rye  or  corn. 

48  1  bushel  of  barley. 

32  1  bushel  of  oats. 

Measures. 

Art,  42, — Cloth  Measure  is  used  in  measuring  cloths, 
and  other  goods  sold  by  the  yard. 
TABLE. 

234  inches  (in.)  make 1  nail.        (na.) 

4  nails  1  quarter,   (qr.) 

4  quarters  1  yard.        (yd.) 

1  qr.=>£  yd,  2  qr.=>£  yd,  3  qr.=%  yd. 


TABLES  OF  MEASURE.  79 

Also,  5  quarters  make  ____  1  Ell  English.     (E.  E.) 

3  quarters  ----  1  Ell  Flemish     (E.  Fl.) 

6  quarters  ----  1  Ell  French.      (E.  F.  not  used.) 

Art.  43.  —  Long  Measure  is  used  in  measuring  distances, 
or  lines  extended  in  length,  breadth,  heighth  and  depth. 

TABLE. 
12  inches  (in.)  make     ..........  1  foot.  (ft.) 

3  feet  ..........  1  yard.  (yd.) 

5£  yds.,  or  16^  ft  ...........  1  rod  or  pole,  (rd.) 

40  rods  ........  .  .  1  furlong.        (fur.) 

8  furlongs  ..........  1  mile.  (m.) 

3  miles  ..........  1  league.         (lea.) 


^degrees  .    ....'.  ..... 

Also  6  feet  make  1  fathom,  (used  in  measuring  deep  water.) 
160  rods  make  £  mile.     80  rods  ^  mile. 

Art.  44.  —  Surveyors'    Measure  is  used  in  measuring 
roads  and  boundaries  of  land,  &c.,  with  chains. 

TABLE. 
7^  inches  make  .............  1  link.  (li.) 

25  links  .............  1  rod  or  pole.    (P.) 

4  poles  .............  1  chain.  (ch.) 

10  chains  .............  1  furlong.          (fur.) 

8  furlongs,  or  80  chains  .............  1  mile.  (m.) 

Art,  45.  —  Square  Measure  is  used  in  measuring  sur- 
faces or  areas,  as  land,  floors,  &c.,  in  which  both  length 
and  breadth  are  considered. 

TABLE. 
144  square  inches  (sq.  in.)  make  ----  1  sq.  foot.  (sq.  ft.) 

9  square  feet  ----  1  sq.  yard.  (sq.  yd.  ) 

301  Sq.  yards,  or  272^  sq.  ft  .....  1  sq.  rod  or  pole.  (sq.  rd.) 

40  sq.  rods  ----  1  rood.  (B.) 

4  roods,  or  160  rods  ----  1  acre.  (A.) 

640  acres  ----  1  sq.  mile.  (sq.  m.) 


80  COMPOUND  NUMBERS. 

This  measure  is  directly  applicable  only  to  surfaces  whose  contents 
are  known,  as  any  number  of  acres,  square  rods,  &c. 

Such  surfaces,  depending  on  the  length  of  certain  lines,  are  found 
by  first  using  long  measure.  Squares  and  rectangles  are  included 
among  them. 

A  Square  is  a  surface  having  four  equal  sides,  and  four 
equal  angles,  which,  also,  are  right  angles. 

If  the  sides  of  a  square  are  each  one  inch  in  length  (long  measure) 
it  is  a  square  inch  ;  if  the  sides  are  each  one  foot,  yard,  rod  or  mile, 
it  is  a  square  foot,  yard,  rod  or  mile. 

If  any  square  is  divided  into  square  feet,   as  in  the    i    i    i    ' i 
annexed  diagram,   it  contains  as  many  rows  of  square    \ 
feet  as  there  are  linear  feet  in  one  side,  and  the  same    1 
number  of  square  feet  in  each  row.     Therefore  if  the    I 


number  of  feet  in  each  row  be  multiplied  by  the  number 
of  rows,  or,  (which  is  the  same,)  if  the  number  of  feet  in  one  side  be 
multiplied  by  itself,  the  product  will  be  the  number  of  square  feet  in 
the  whole  square.  The  same  is  true  of  yards,  rods,  &c.  Hence  the 

RULE. — To  find  the  contents  or  area  of  a  square,  multiply 
the  length  of  any  one  of  its  sides  by  itself. 

There  is  no  difference  between  a  square  foot  and  a  foot  square,  but 
there  is  a  difference  between  such  expressions  as  3  square  feet,  and 
3  feet  square.  A  square  yard  is  3  feet  square,  and  contains  9  square 
feet,  (see  last  diagram,)  hence  the  difference  between  3  feet  square 
and  3  square  feet  is  6  square  feet.  The  same  is  true  of  rods  square 
and  square  rods,  &c. 

A  Rectangle  is  like  a  square,  only  it  is  longer  than  it 
is  wide,  and  its  contents  are  found  by  multiplying  its 
length  by  its  breadth,  both  being  reduced  if  necessary  to 
the  same  name. 

This  diagram  represents  a  rectangle  6  ft.  long 
and  3  ft.  wide— contents  6X3=18,  the  number 
of  square  feet. 

Since,  also,  the  product  of  two  numbers  di- 
vided by  one  of  them  gives  the  other,   either 
side  of  a  rectangle  may  be  found  by  dividing  the  contents  by  the 
other  side  ;  as,  18  sq.  ft.  -H*  ft.  (wide)=6  ft.  long. 

Art,  46,— Cubic  Measure  is  used  in  measuring  solids, 
in  which  length,  breadth  and  thickness  are  considered. 
It  is  sometimes  called  Solid  Measure. 


COMPOUND  NUMBERS.  81 

TABLE. 

1728  cubic  inches  (CM.  in.)  make  1  cubic  foot  (cu,  ft.) 
27  cubic  feet  ..........  1  cubic  yard  (cu.  yd.) 

40  feet  of  round  or  ) 

50  feet  of  hewn      [timt>er       1  ton  (T.) 

16  cubic  feet  ..........  1  cord  foot  (0.  ft.) 

8  cord  feet  or        ) 
128  cubic  feet  }  ..........  l  cord  of  wood' 

A  Cube  is  a  solid  having  six  equal  and  square  sides. 
If  the  sides  are  each  a  square  inch,  the  solid  is  a 
cubic  or  solid  inch  ;  if  the  sides  are  each  a  square  foot  or 
yard,  the  solid  is  a  cubic  or  solid  foot  or  yard,  &c. 

If  the  base  of  a  cube  be  4  feet  square,  it  will  con- 
tain 16  square  feet,  and  a  solid  on  it,  1  foot  high, 
would  contain  16  solid  feet,  2  feet  high  16X2= 
32  solid  feet;  four  feet  high  16X4=64  solid 
feet  ;  hence 

RULE.  —  The  contents  of  a  cube,  or  any  regular  solid,  may 
be  found  by  multiplying  together  the  length,  breadth  and 
thickness. 

The  contents  and  any  two  of  the  dimensions  being  given,  the  other 
dimension  may  be  found  by  dividing  the  contents  by  the  product  of 
the  given  dimensions. 

Art.  47.  —  Wine  Measure  is  used  in  measuring  liquids, 
except  beer  and  ale. 

TABLE. 


2    pints 

1  Quart         (ot  ) 

4    quarts 

1  erallon        (e*al  } 

314r  gallons 

.     I  barrel         (bbl  ) 

63    gallons  (or  2  barrels)  .  .  . 

.  .  .1  hogshead  (hhd  ) 

2    hogsheads                  .  .  . 

.  .  1  pipe           (D  } 

2    pipes  (or  4  hhds.) 

.  .1  tun            (T.) 

Also  42    fallons                        •  •  • 

.  .  .1  tierce        (tr  ) 

48   eallons 

.  .1  puncheon  (  pun.) 

A  gallon  contains  231  cubic  inches 

4* 


82  COMPOUND  NUMBERS. 

Art*  48. — Beer   Measure  is  used  for  measuring  beer, 
and  formerly  milk. 

TABLE. 

2  pints    make 1  quart. 

4  quarts  1  gallon. 

36  gallons  1  barrel. 

5i  gallons  1  hogshead. 

A  beer  gallon  contains  282  cubic  inches. 

Art.  49. — Dry  Measure  is  used  for  measuring  grain, 
fruit,  vegetables,  &c. 

TABLE. 

2  pints  (pt.)  make 1  quart         (qt.) 

8  quarts  1  peck  (pk.) 

4  pecks  1  bushel       (bu.) 

Also,    8  bushels       make 1  quarter      (q.) 

36  bushels  1  chaldron. 

32  bushels  1  chaldron  (in  U.  S.) 

A  bushel  in  form  of  a  cylinder  is  18 1  in.  in  diameter  and  8  inches 
deep.  It  contains  2150|^  cubic  inches. 

Art.  50.— Time  Measure  is  used  in  making  divisions 
of  time. 

TABLE. 

60  seconds  (sec.)  make 1  minute  (in.) 

60  minutes  1  hour  (h.) 

24  hours  1  day  (and night)  (d.) 

7  days  1  week  (w.) 

4  weeks  or  ) 

1  month  (mo.) 

30  days  i 

12  months  (calendar)  or  i 

13  "      (lunar)  or  (365d.nearly)  f   lvear-  GrO 

The  Solar  year  consists  of  365  days,  6  hours  (very  nearly)  in  which 
the  earth  makes  one  revolution  around  the  sun. 

The  Civil  year  is  365  days.  This  makes  a  difference  of  one  day  in 
four  years,  which  is  added  to  the  month  of  February,  making  it  con- 


COMPOUND   NUMBERS. 


83 


sist  of  29  instead  of  28  days.     This  occurs  whenever  the  number  of 

years  is  divisible  by  4,  which  is  Leap  Year 

The  addition  of  one  day  every  Leap  Year,  makes  about  one  day 

too  much  in  100  yeJars,  hence  it  is  not  added  during  the  year  which 

completes  a  century,  as  A.D.  1800,  A.D.  1900. 

The  names  of  the  months  and  the  number  of  days  in  each,  are 

as  follows  : 

January       (2d    Winter  month,)   31  days. 

February       3d         "  28  or  29  days. 

March  1st  Spring  31  days. 

April  2d      «  30 

May  3d       "  31 

June  1st  Summer  30 

July  2d        <k  31 

August          3d         "  31 

September    1st  Fall  30 

October         2d     "  31 

November     3d     "  30 

December     1st  Winter  31 

To  assist  in  remembering  the  number  of  days  in  each  month  the 

following  lines  have  been  much  used  : 

Thirty  days  hath  September, 
April,  June,  and  November, 
February  twenty-eight  alone 
All  the  rest  have  thirty-one  ; 
Except  in  Leap  Year,  then  is  the  time 
When  February  has  twenty-nine. 

TABLE, 

THE  NUMBER  OF  DAYS  FKOM  ANY  DAY  OF  ONE  MONTH  TO  THE  SAME 
DAY  OF  ANY  OTHER  MONTH  IN  THE  SAME  YEAR. 


FROM 
ANY  DAY 
OF 

TO  THE  SAME  DAY  OF 

JAN. 

FEB. 

MAE. 

APL. 

MAY. 

J'NE. 

J'LY. 

AUG. 

SEP. 

OCT. 

NOV. 

DEC. 

JANUA'Y 

367 

31 

59 

90 

120 

151 

181 

212 

243 

273 

304 

334 

FEBR'Y  . 

334 

365 

28 

59 

89 

120 

150 

181 

212 

242 

273 

303 

MARCH. 

306 

337 

365 

31 

61 

92 

122 

153 

184 

214 

245 

275 

APBZL.. 

275 

306 

334 

365 

30 

61 

91 

122 

153 

183 

214 

244 

MAT... 

245 

276 

304 

335 

365 

31 

61 

92 

123 

153 

184 

214 

JUNE  .  . 

214 

245 

273 

304 

344 

365 

30 

61 

92 

122 

153 

183 

JULY  .  . 

184 

215 

243 

274 

304 

335 

365 

31 

62 

92 

123 

153 

AUGUST 

153 

184 

212 

243 

273 

304 

334 

365 

31 

62 

92 

122 

SEPTM'K 

122 

153 

181 

212 

242 

273 

303 

334 

365 

30 

61 

91 

OCTOB'R 

92 

123 

151 

182 

212 

243 

273 

304 

335 

365 

31 

61 

NOVEM. 

61 

92 

120 

151 

181 

212 

242 

273 

304 

334 

365 

30 

DECEM. 

31 

62 

90 

121 

151 

182 

212 

243 

274 

304 

335 

365 

The  number  of  days  is  opposite  the  first  given  month  and  under 
the  other. 


84  COMPOUND  NUMBIiKS. 

Art.  51. — Circular  Measure  is  used  in  measuring  cir- 
cles, as  in  finding  the  latitude  and  longitude  of  places  or 
of  heavenly  bodies. 

A  Circle  is  a  figure  bounded  by  a  curved 
line,  in  which  every  point  is  equally  dis- 
tant from  the  centre. 

The  Circumference  is  the  curved  line. 
The  Diameter  is  the  straight  line  passing 
through  the  centre  to  opposite  points  in  the  circumfer- 
ence. A  Radius  is  any  straight  line  from  the  centre  to 
the  circumference. 

Circles  are  divided  into  360  parts  called  degrees,  vary- 
ing according  to  the  size  of  the  circles. 

TABLE. 

60  seconds  (")  make 1  minute  (') 

60  minutes  1  degree  (°) 

30  degrees  1  sign  (S.) 

12  signs  or  360°        1  circle  (C.) 

Art.  52. — MISCELLANEOUS  TABLE. 

12  units  or  things  make 1  dozen. 

12  dozen  1  gross. 

12  gross  1  great  gross. 

20  units  or  things 1  score. 

PAPEB. 

24  sheets  make  1  quire. 

20  quires  1  ream. 

BOOKS. 

A  sheet  folded  in  two  leaves  makes  a  folio. 

"        "  four  "          "      a  quarto  or  4to. 

"        "  eight  "         "      an  octavo  or  8vo. 

"        "  twelve  "         "     a  duodecimo  or  12mo. 

"        "  eighteen       "          "      an  ISrno. 

"        "  twenty-four"         "     a  24mo. 


COMPOUND   NUMBERS.  85 


Reduction, 

Art.  53.—  Reduction  of  Compound  Numbers  is  chang- 
ing one  denomination  or  name  to  another,  without  alter- 
ing its  value,  and  is  either  descending  or  ascending. 

Art.  54,  —  Beduction  descending  is  changing  a  greater 
denomination  to  a  less,  as  pounds  to  shillings,  &c. 

Reduction  ascending  is  changing  a  less  denomination 
to  a  greater,  as  farthings  to  pence,  &c. 

Examples  illustrating  reduction  descending  : 

EXAMPLE  1.  —  Reduce  £24  15s.  10)£d.  to  farthings. 

Process.  —  Reduction  descending  because  it  is  £  s.  d.  f. 

changing  greater  denominations  (pounds,  &c.)  to  24  15  10  2 

a  less  (farthings.)  _20 

1.  Change  the  pounds  to  shillings.    Since  there  £95  ghJU'gg 
are  20s.  in  £1,  there  are  in  £24,  20  times  as  many  j2 
shillings,  which  are  480s.,  to  which  add  the  15s. 

and  the  sum  will  be  495s.  595J  pence. 

2.  Change  the  shillings  to  pence.     Since  there 

are  12d.  in  Is.,  there  are  in  495s.  12  times  as  many  Ans.  23802  fart'gs. 
pence,  which  are  5940d.,  to  which  add  the  10  jd. 
and  the  sum  will  be  5950£  or  5950d.  2far. 

3.  Change  the  pence  to  farthings.     Since  there  are  4  far.  in  Id., 
there  are  in  5950d.  4  times  as  many  farthings,  which  are  23800far.,  to 
which  add  the  2far.  and  the  sum  will  be  23802far.,  the  answer. 

Ex.  2.  In  15  rods,  4  yds.,  2  ft.,  9  in.,  how  many  inches  ? 

Process.—  Eeduction  descending,  as  in  the  rds.  yds.  ft.  in. 

preceding  example.  15      4       2       9 

1.  Change  the  rods  to  yards.     Since  there  5k 
are  5  5  yards  in  one  rod,  in  15  rods  there  are  ~~y£ 
5s*  times  as  many  yards,  which  are  82  \  yds.,  75 
to  which  add  the  4  yds.,   and  the  sum  will 

be  86|  yds.  8^  7**> 

2.  Change  the  yards  thus  found  into  feet.  rf 
Since  there  are  3  feet  in  1  yard,  in  865  yds.               261  i  ft. 
there  are  3  times  as  many  feet,  which  are                 12 
259  2  ft,  to  which  add  the  2  ft.,  and  the  sum    ^m  ~3i47~in 
will  be  261i  ft. 

3.  Change  the  feet  to  inches.     Since  there  are  12  inches  in  1  foot, 
in  261  i  ft.,  there  are  12  times  as  many  inches,  which  are   3138  in., 
to  which  add  the  9  in.  ,  and  the  sum  will  be  3147  inches,  the  answer. 

*  Since  this  is  by  a  rule  in  Fractions,  the  teacher  should  explain  the  process  in 
such  cases,  as  it  is  explained  in  Fractions. 


86  COMPOUND  NUMBERS. 

RULE. — In  reduction  descending,  multiply  the  greatest  de- 
nomination by  that  number  which  it  takes  of  the  next  less  to 
make  one  of  this  greater,  and  add  to  the  product  any  given 
number  of  the  same  name. 

Proceed  thus  till  the  required  denomination  is  found. 

EXAMPLES  ILLUSTRATING  REDUCTION  ASCENDING. 

Ex.  3.  Reduce  23802  farthings  to  pounds,  &c. 

Process.— Keduction  ascending,  changing  4)23802  far. 

a  less  denomination  (farthings)   to  greater  12)5950  d  -4-2far 
(pounds,  &c.) 

1.  Change  the  farthings  to  pence.    Since  20)49.5s.+10. 

4  farthings  make  1  penny,  there   are  as  24-f-15s.-{-10£d. 

many  pence  in  23802  far.  as  there  are  times 

4  farthings,  which  are  5950  times,  and  2  farthings  remaining.    There- 
fore 23802far.  are  5950d.  2far. 

2.  Change  the  pence  thus  found  to  shillings.     Since  12  pence  make 
1  shilling,  there  are  as  many  shillings  in  5950d.  as  there  are  times  12d. 
which  are  495  times,  and  lOd.  rem.     Therefore  5950d.  are  495s.  lOd . 

3.  Change  the  shillings  to  pounds.     Since  there  are  20  shillings  in 
1  pound,  there  are  as  many  pounds  in  495s.   as  there  are  times  20s. 
which  are  24  times,  and  15s.  rem.     Therefore  495s.  are  £24  15s. 

Also  23802far.  are  £24  15s.  10£d. 

Ex.  4.  In  3147  inches,  how  many  rods,  &c. 

Process. — Keduction  ascending  as  in  the    12)3147  in. 
preceding  example.  3)262  ft.+3  in. 

1.  Change  the  inches  to  feet.     Since  12 

in.  make  1  ft.,  there  are  as  many  feet  in  5*)8J  yds.+  Lft. 

3147  in.  as  there  are  times  12  in.,  which 

are  262  times,  and  3  in.  rem.     Therefore  11)174 

3147  in.  are  262  ft.  3  in.  T5  rds.+9  half  yds. 

2.  Change  the  feet  thus  found  to  yards.  /^  y(jg  ^  ft  g  jn  \ 
Since  3  ft-  make  1  yd.,  there  are  as  many  Ans,  15  ^8,4  yds.  2  ft.  9in. 
yds.  in  262  ft.   as  there  are  times  3  ft., 

which  are  87  times.     Therefore  262  ft.  are  87  yds.  1  ft. 

3.  Change  the  yards  to  rods.     Since  5£  yds.  make  1  rod,  there  are 
as  many  rods  in  87  yds.   as  there  are  times  5£  yds.  which  are  15 
times,  *  and  4|  yds.  (=4  yds.  1  ft.  6  in. )  rem.     Therefore   87  yards 
are  15  rods,  4  yds.  1  ft.  6  in. 

And  3147  in.  are  15  rods,  4  yds.  2  ft.  9  in. 

*  By  a  rule  in  Fractions. 


ENGLISH  MONEY.  87 

KULE. — In  reduction  ascending,  divide  the  given  denomi- 
nation by  that  number  which  it  takes  of  this  denomination  to 
make  one  of  the  next  less. 

Proceed  thus  until  the  greatest  denomination  required  is 
found,  which,  with  the  remainders,  in  regular  order,  will  be 
the  answer. 

Art,  55.— English  or  Sterling  Money, 

MENTAL     EXERCISES. 

How  many  farthings  in  3  pence  ? 

Process. — Since  there  are  4far.  in  1  penny,  in  3d.  there  are  4  times 
as  many  farthings  as  pence,  which  are  12  farthings  ;  or  there  are 
4  times  3  far. 

How  many  farthings  in  5d  ?  7  ?  9  ?  11  ?  4  ?  6  ?  8? 
10  ?  12  ?  in  1  shilling  V  1  pound  ? 

In  50  farthings  how  many  pence  ? 

Process. — Since  4  farthings  make  1  penny,  in  50  farthings  there 
are  as  many  pence  as  there  are  times  4  farthings,  which  are  12  times, 
and  2  farthings  remaining.  Therefore  50  far.  are  12d.  2far. 

How  many  pence  in  8  farthings  ?  16  ?  24  ?  32  ?  40  ? 
48?  10?  20?  25?  30?  42?  50? 

How  many  pence  in  2  shillings  ?  4?  6?  8?  10?  5? 
7?  9?  12?  in  1  pound? 

How  many  shillings  in  24  pence?  48?  72?  36?  60? 
96?  30?  40?  56?  63?  81?  100?  144? 

How  many  shillings  in  2  pounds  ?  4?  6?  9?  3?  5? 
7?  10? 

How  many  pounds  in  40  shillings  ?  60  ?  80  ?  50  ?  65  ? 
72?  84?  100?  120? 

How  many  shillings  in  2  guineas  ?  4?  6?  3?  5?  7? 
10? 

How  many  guineas  in  42  shillings  ?  63  ?  84  ?  50  ?  65  ? 
76? 

How  many  farthings  in  1  shilling  ?    2  ?    3  ? 

How  many  pence  in  1  pound  ?    2  ?    3  ? 

How  many  shillings  in  48  far.  ?    96  ? 


88 


COMPOUND  NUMBEES. 


EXAMPLES   FOR   THE   SLATE,    ETC. 


Reduce — 

(5.)  £17  2s.  6d.  to  pence. 

(6.)  £11  15s.  4d.  to  farthings. 

(7.)  £15  18s.  lOd.  to  pence. 

(8.)  £24  12s.  8d.  to  far. 

(9.)  £45  8s.  7>£d.  to  far. 
(10.)  £8  4s.  to  pence. 
(11.)  10s.  9d.  Sfartofar. 
(12.)  £1813s.  8£d.  to  far. 
(18.)  £12  14s.  to  shillings. 
(14.)  lid.  2far.  to  farthings. 
(15.)  £4  Gd.  to  farthings. 
(16.)  £5  15s.  to  pence. 
(17.)  10s.  3far.  to  far. 
(18.)  £10  0£d.  to  far. 

Reduce — 

(33.)  £1510s.  9d.  to  pence. 
(34.)  23470far.  to  pence. 
(35.)  18s.  7d.  3far.  to  far. 
(36.)  1340d.  to  pounds. 


(19.)  23475far.  to  pounds,  &c. 
(20.)  3756d.  to  pounds,  &c. 
(21.)  7620s.  to  pounds,  &c. 
(22.)  4325far.  to  shillings,  &c. 
(23.)  2754d.  toshiUings,  &c. 
(24.)  3240far.  to  pence,  &c. 
(25.)  4360s.  to  pounds,  &c. 
(26.)  1345d  to  pounds,  &c. 
(27.)  7563far.  toshiUings,  &c. 
(28.)  2560id.  to  pounds. 
(29.)  1781s.  to  pounds,  &c. 
(30.)  156M.  to  shillings,  &c. 
(31.)  1682far.  to  pence.  &c. 
(32.)  5146|d.  to  pounds,  &c. 

(37.)  12560far.  to  pounds,  &c. 
(38. )  £16  4s.  to  pence. 
(39.)  1260s.  to  pounds. 
(40.)  17s.  6^d.  to  far.  &c. 


Art.  56.— Troy  Weight. 

MENTAL   EXERCISES. 

How  many  grains  in  2  pennyweights  ?  4  ?  6?  3?  5? 
7?  9?  10? 

How  many  pennyweights  in  48  grains?  72?  96?  30? 
56?  81? 

How  many  pennyweights  in  2  ounces  ?  4?  6?  3?  5? 
8?  10?  7?  12? 

How  many  ounces  in  40  pennyweights  ?  60  ?  100  ?  50  ? 
75  ?  110  ? 


COMPOUND   NUMBERS.  89 

How  many  ounces  in  2  pounds  ?  4?  6?  8?  3?  7? 
10?  5?  9?  12? 

How  many  pounds  in  24  ounces  ?  36  ?  48  ?  72  ?  96  ? 
144?  30?  56?  68?  112? 

EXAMPLES  FOB  THE   SLATE. 

Reduce — 


(41.)  59125  gr.  to  oz. 

(42.)  lOlbs.  10  oz.  lOpwts.  to 

pwts. 

(43.)  6743  pwts.  to  Ibs. 
(44.)  1000  Ibs.  to  pwts. 
(45.)  1000  pwts.  tooz. 


(46.)  62  Ibs.  5oz.  to  pwts. 

(47.)  2840  grs.  to  oz. 

(48.)  2840 Ibs.  tooz. 

(49.)  12  Ibs.  15  pwts.  to  grains. 

(50.)  2895  pwts.  to  Ibs. 


Art.  57.— Avoirdupois  Weight. 

MENTAL   EXERCISES. 

How  many  drams  in  2  ounces  ?4?6?8?3?5?7?9?10? 

How  many  ounces  in  32  drams  ?  48  ?  64  ?  36  ?  50  ?  75  ? 

How  many  ounces  in  2  pounds  ?  4?6?8?3?  5?7?  10? 
9?  11? 

How  many  pounds  in  32  ounces  ?  48  ?  64  ?  30  ?  54  ?  75  ? 

How  many  pounds  in  2  quarters  ?4?3?8?5?10? 

How  many  quarters  in  50  pounds  V  75  ?  100  ?  40  ?  80  ?  108  ? 

How  many  quarters  in  2  hundred  weight  ?3?5?7?4?6? 
9?  12? 

How  many  hundred  weight  in  8  quarters  ?  16  ?  24  ?  12  ? 
20?  10?  15?  25? 

How  many  hundred  weight  in  2  tons?  4?  6?  3?  5?  7?  10? 

How  many  tons  in  40  hundred  weight  ?  60  ?  80  ?  100  ? 
50?  75? 

EXAMPLES   FOB   THE   SLATE. 

Reduce — 


(51. )  3  cwt.  to  pounds. 
(52.)  4815  Ibs.  to  cwts. 
(53.)  2748000  dr.  to  tons. 
(54.)  4  T.  15  Ibs.  to  pounds. 


(55.)  67200  oz.  to  cwt. 

(56.)  15  cwfc.  1  qr.  12  oz.  to  oz. 

(57.)  22500  Ibs.  to  tons. 

(58.)  3  T.  15  cwt.  16  Ibs.  to  Ibs. 


90  COMPOUND   NUMBERS. 


(59.)  4  cwt.  3  qrs.  to  pounds. 

(60.)  1120  oz.  to  pounds. 

{61.)  1225  Ibs.  to  drams. 

(62.)  1750  Ibs.  tocwts. 


(63.)  2000  Ibs.  to  ounces. 
(64.)  4000  Ibs.  tocwts. 
(65.)  28000  Ibs.  to  tons. 


Art.  58,— Apothecaries'  Weight. 

MENTAL  EXERCISES. 

How  many  grains  in  2  scruples  ?  4  ?  6  ?  3  ?  5?8?7?  9?  10? 
How  many  scruples  in  40  grains  ?  60?  100?  50?  65?  87? 
112? 

How  many  scruples  in  3  drams  ?  5?  7?  2  ?  4  ?  6  ?  9  ?  11  ? 
How  many  drams  in  2  ounces  ?4?  6?  3?  7?  9?  8?  10?  12? 
How  many  ounces  in  3  pounds  ?  2?  4?  6?5?7?10?  9?  12? 

EXAMPLES  FOB  THE   SLATE. 

Reduce — 


(66.)  47230  gr.  to  ounces. 
(67.)  5375  Ibs.  to  ounces. 
(68)  74376  dr.  to  pounds. 
(69.)  10752  gr.  to  ounces. 


(70.)  1728  Ibs.  to  ounces. 
(71.)  1730  dr.  to  ounces. 
(72.)  6  Ib.  4  oz.  to  drams. 
(73.)  5000  gr.  to  ounces,  &c. 


Art,  59, — Cloth  Measure. 

MENTAL  EXERCISES. 

How  many  inches  in  2  nails  ?  4?   6?  3?  5?   7?   10?  12? 
How  many  nails  in  5  inches  ?   10  ?   15  ?  20  ?  6  ?   12  ? 
How  many  nails  in  2  quarters  ?  4?  3?   5?  7?  6?  8?  11? 
How  many  quarters  in  3  yards  ?   5?  8?   10?   12? 
How  many  yards  in  8  quarters  ?  16  ?  20  ?  12  ?  9  ?  15  ?  18  ? 
21?   24? 

EXAMPLES  FOR  THE   SLATE. 

Reduce — 

(74.)  128yds.  to  E.  E. 
(75.)  75yds.  to  E.  Fl. 
(76.)  764  qrs.  to  yds. 
(77.)  10  yds.  3  qrs.  to  nails. 


(78.)  11^  yds.  to  E.  E. 

(79.)  16>£  yds.  to  quarters. 

(80.)  20  yds.  3  qrs.  to  inched. 

(81.)  6  E.  E.  to  E.  F. 


COMPOUND  NUMBERS.  91 

Art.  60. — Long  Measure, 

MENTAL   EXERCISES. 

How  many  inches  in  3  ft.?  5?  8?  7?  9?   11?   12? 

How  many  feet  in  24  inches  ?  36?  48?  60?  30?  56?  63? 
72?  108?  144? 

How  many  feet  in  3  yards  ?2?4?6?5?7?9?10?12? 

How  many  yards  in  6  ft.?  9?  15?  18?  24? 

How  many  yards  in  3  rods  ?   6?  5?   7?  10? 

How  many  feet  in  2  rods  ?  4  ?   3  ?   5  ? 

How  many  rods  in  3  furlongs  ?  4  ?  6  ?  5  ? 

How  many  miles  in  16  furlongs  ?  24  ?  40  ?  32  ?  56  ?  72  ?  64  ? 
10?  20?  25? 

EXAMPLES  FOR  THE   SLATE. 

Beduce — 


(82.)  103  in.  to  yards. 

(83.)  7  yds.  10  in.  to  inches. 

(84.)  3840  rods  to  miles. 


(85. )  7  M.  6  fur.  30  rods  to  rods. 
(86.)  2910  rds.  to  miles. 
(87.)  5  yds.  1  in.  to  inches. 


Art.  61.— Square  Measure. 

MENTAL   EXERCISES. 

How  many  square  feet  in  2  square  yards  ?  4?  6?  8?  3? 
5  ?  7  ?  10  ?  12  ? 

How  many  square  yards  in  18  square  feet  ?  27  ?  54  ?  36  ? 
63?  99?  20?  44?  56? 

How  many  square  feet  on  a  board  1  ft.  wide  and  12  ft.  long 
(12  ft.  by  1  ft.  ?)  in  a  door  6  ft.  high  and  3  ft.  wide  (6  ft.  by  3  ft.  ?) 
in  a  room  12  ft.  by  11  ft.  ? 

What  is  the  length  of  a  board  2  ft.  wide  and  containing  24 
square  feet  ?  a  pane  of  glass  9  inches  wide  and  containing  108 
square  inches  ?  of  a  room  12  feet  wide  and  containing  180 
square  feet  ?  of  a  field  10  rods  wide  and  containing  1  acre  ? 

What  is  the  difference  between  4  square  feet  and  4  feet 
square  ?  5  square  rods  and  5  rods  square  ? 


92  COMPOUND   NUMBERS. 

EXAMPLES  EOR  THE  STjATE. 


Reduce — 

(88.)  243  sq.  rods,  to  acres,  &c. 
(89.)  8  sq.  yds  to  sq.  in. 
(90.)  4176  sq.  in.  to  sq.  yds. 
(91.)  24000  sq.  rds.  to  acres. 


(92.)  1  sq.  M.  to  sq.  rods. 

(93.)  71680  sq.  rds.  to  acres. 

(94. )  16  A.  18  sq.  rds.  to  sq.  rds. 

(95.)  78436  ft.  to  acres. 


96.  How  many  square  inches  in  a  window  pane  15  in.  by 
12  in.? 

97.  How  many  sq.  ft.  in  a  looking  glass  36  in.  long  and  24  in. 
wide  (36  in.  by  24  in.  ?) 

98.  How  many  sq.  ft.  in  a  floor  12  ft.  by  10  ft.  ? 

99.  How  many  sq.  yds.  in  a  floor  18  ft.  by  15  ft.  ? 

100.  How  many  sq.  rods  in  a  field  40  rds.  by  20  rds.  ? 

101.  How  many  acres  in  a  field  45  rds.  by  30  rds.  ? 

102.  How  many  acres  in  a  farm  ^  M.  by  60  rds.  ? 

103.  What  is  the  length  of  a  window  pane  15  in.  wide  and 
containing  300  square  inches  ? 

104.  What  is  the  width  of  a  looking  glass  60  in.  long  and 
containing  15  square  feet  ? 

105.  What  is  the  length  of  a  room  12  ft.  wide  and  contain- 
ing 192  square  feet  ? 

106.  What  is  the  width  of  a  room  which  requires  30  sq.  yds. 
of  carpeting,  and  is  18  feet  long  ? 

107.  How  many  sq.  yds.  in  6  M.  7  sq.  rods  ? 

108.  How  many  square  miles  in  6400  acres  ? 

109.  How  many  sq.  yds.  of  paper  will  cover  the  walls  of  a 
room  24  feet  by  20,  and  12  feet  high  ;  and  how  many  square 
yards  of  ceiling  are  there  ? 

110.  How  many  square  miles  in  92160  acres  ? 

111.  How  many  yards  of  muslin  3  qrs.  wide,  will  line  a  quilt 
8  feet  square  ? 


Art.  62,— Cubic  Measure. 

Reduce — 

112.  4  cord,  16  cub.  feet  to  cub.  feet. 

113.  1  cub.  yd.,  10  cub.  feet  to  cub.  inches. 

114.  34  cords,  64  cub.  feet  to  feet. 


COMPOUND  NUMB 


115.  31104  cub.  inches  to  cub.  feet. 

116.  3584  cub.  feet  to  cords. 

117.  442368  cub.  inches  to  cords. 


118.  How  many  cubic  feet  of  wood  in  a  load  of  wood  8  ft. 
long,  4  ft.  wide,  and  5  ft.  high  ? 

119.  How  many  cubic  yards  in  a  cellar  24  ft.  long,  15  ft. 
wide,  and  6  ft.  deep  ? 

120.  How  many  cords  of  wood  in  a  pile  40  ft.  long,  8  ft. 
wide,  and  6  ft.  high  ? 

121.  How  many  bricks  will  it  take  to  lay  the  foundation  of  a 
house  32  ft.  by  30,  the  height  of   the  foundation  being  8  ft. 
and  the  thickness  1  ft. ;  the  bricks  being  8  in.  long,  4  in.  wide, 
and  2  in.  thick  ? 

Art.  63,— Wine  Measure. 

MENTAL  EXERCISES. 

How  many  gills  in  3  pints  ?    5  ?    7  ?    10  ? 
How  many  pints  in  8  gills  ?    16  ?    20  ?    32  ? 
How  many  pints  in  2  quarts  ?    4?     6?     8?     12? 
How  many  quarts  in  3  gallons  ?    5?    7?    9?    12? 
How  many  quarts  in  4  pints  ?    8  ?    10  ?    16  ?    22  ? 
How  many  gallons  in  8  quarts  ?     12  ?    20  ?    25  ? 
How  many  gallons  in  2  barrels  ?     2  hogsheads  ? 

EXAMPLES  FOB  THE  SLATE. 

Reduce — 


(122.)  2  hhd.  2  qt.  1  pt.  to  gills. 
(123.)  17  gaL  2  qts.  to  pints. 


(124.)  140  pts.  to  gallons,  &c. 
(125.)  40736  gills  to  tuns,  &c. 


Reduce  by  Beer  Measure — 
(126.)  3  bbls.  16  gals,  to  pts.     |  (127.)  1730  pts.  to  barrels,  &c. 

128.  How  many  gills  in  1  hogshead  of  wine  ? 

129.  How  many  pints  in  1  barrel  of  beer  ? 

130.  How  many  quarts  in  5  hhds.,  31  }£  gals,  of  vinegar  ? 


94  COMPOUND  NUMBEKS. 

131.  How  many  quarts  in  3  hhds.  of  ale  ? 

132.  How  many  hhds.,  &c.,  in  1200  gals.  ? 

133.  How  many  barrels  in  850  gals,  of  beer  ? 


Art.  64. — Dry  Measure. 

MENTAL    EXEKCISES. 

How  many  pints  in  4  quarts  ?    6?    9?    12?    15? 
How  many  quarts  in  4  pints  ?    6?    12?    9?    15? 
How  many  pecks  in  16  quarts  ?    24  ?    40  ?    56  ? 
How  many  pecks  in  2  bushels  ?    5  ?    3  ?    7  ? 
How  many  bushels  in  16  pecks  ?    32  ?    48  ?    72  ? 

EXAMPLES  FOB  THE   SLATE. 

Reduce — 


(134.)  12  bus.  3pks.  to  qts. 

(135.)  3  pks.  1  pt.  to  pints. 

(136.)  15  bu.  6  qts.  to  qts. 

(137.)  4^  pks.  to  pints. 


(138.)  1000  quarts  to  bushels. 

(139,)  560  pecks  to  bushels. 

(140.)  32140  pints  to  pecks. 

(141.)  2764  qts.  to  bushels. 


142.  In  100  bushels  how  many  quarts  ? 

143.  In  1200  pints  how  many  pecks  ? 


Art.  65.— Time. 

MENTAL   EXEKCISES. 

How  many  seconds  in  2  minutes  ?    4  ?    6  ? 
How  many  minutes  in  120  seconds  ?    180  ? 
How  many  minutes  in  3  hours  ?    5  ?    7  ?    10  ? 
How  many  days  in  2  weeks  ?    4  ?    7  ?    12  ? 
How  many  weeks  in  3  months  ?    5  ?    7  ?    10  ? 
How  many  years  in  24  months  ?    36  ?    48  ?    65  ? 

EXAMPLES  FOB  THE   SLATE,    ETC. 

Reduce — 

144.  2  hours,  3  minutes,  50  seconds,  to  seconds. 

145.  3  days,  30  minutes,  to  minutes. 

146.  1  year,  6  months,  27  days  to  hours. 

147.  3  years,  2  weeks,  10  hours  to  seconds. 


EEDUCTION.  95 

148.  9  months,  15  days,  10  minutes  to  minutes. 

149.  1  solar  year  to  seconds. 

150.  12000  minutes  to  days,  &c. 

151.  99840  seconds  to  days,  &c. 

152.  7200  seconds  to  hours. 

153.  56000  minutes  to  months. 

154.  450000  seconds  to  days. 

155.  12500  days  to  years. 

156.  How  many  hours  in  5  years,  6  months,  3  weeks,  4  days  ? 

157.  How  many  days  in  365000  seconds  ? 

158.  How  many  seconds  in  1  day  ? 


66.— Circular  Measure. 


Reduce— 

159.  10  S.  15°,  45',  30"  to  seconds. 

160.  3000000  seconds  to  signs. 

161.  18°,  36',  12"  to  seconds. 

162.  25000"  to  degrees. 


Art,  67,— Promiscuous  Examples  in  Reduction  of 
Compound  Numbers. 

EXERCISE  I. 

1.  How  many  farthings  in  £5  10s.  6d.  ? 

2.  How  many  pounds  in  3540  pwt.  ? 

3.  How  many  pounds  in  3  T.  7  cwt.  3  qr.  ? 

4.  How  many  pounds  in  7580  scruples.  ? 

5.  How  many  nails  in  25  yd.  3  qr.  ? 

6.  How  many  miles  in  500000  ft.  ? 

7.  How  many  sq.  yards  in  5  A.  1  R.  12  rd.  ? 

8.  How  many  sq.  yards  in  350  000  sq.  in.  ? 

9.  How  many  sq.  inches  in  a  looking  glass  30  in.  long  and 
1 8  in.  wide  ? 


96  REDUCTION. 

10.  How  many  cubic  feet  in  6  C.  72  ft.  ? 

11.  How  many  cubic  feet  in  a  load  of  wood  8  feet  long,  4  feet 
wide,  5  feet  high  ? 

12.  How  many  gallons  in  12548  gills  ? 

13.  How  many  gallons  in  3  T.  2  hhd.  21  gal.  ? 

14.  How  many  quarts  in  9  hhd.  15  gal.  3  qt.  of  beer  ? 

15.  How  many  pints  in  1  bu.  4  qts.  ? 

16.  How  many  days  in  325760  seconds  ? 

17.  How  many  seconds  in  14°  16'  15"  ? 

EXERCISE  n. 

18.  How  many  pounds  in  2540  farthings. 

19.  How  many  pennyweights  in  12  pounds,  9  ounces  ? 

20.  How  many  hundredweight  in  36456  drams  ? 

21.  How  many  grains  in  7  pounds,  8  ounces,  1  scruple  ? 

22.  How  many  yards  in  1650  nails  ? 

23.  How  many  yards  in  5  miles,  3  furlongs,  4  yards  ? 

24.  How  many  acres  in  a  field  69  rods  long,  45  rods  wide  ? 

25.  How  many  cords  of  wood  in  a  pile  5  rods  long,  8  feet  wide 
and  6  feet  high  ? 

26.  How  many  pints  in  5  hhd.  36  gallons,  1  quart,  1  pint  ? 

27.  How  many  barrels  in  5000  quarts  of  beer  ? 

28.  How  many  bushels  in  3540  quarts  ? 

29.  How  many  minutes  in  3  years,  4  months,  1  week,  2  days  ? 

30.  How  many  degrees  in  9863  seconds  of  a  circle  ? 

EXERCISE  m. 

31.  How  many  pence  in  £6  12s.  4d. 

32.  How  many  pounds  in  75603  grains  of  silver  ? 

33.  How  many  pounds  in  5  T.  10  cwt.  20  pounds  ? 

34.  How  many  pounds  in  7856  grains  of  calomel  ? 

35.  How  many  yards  in  6  ells  English  ? 

36.  How  many  yards  in  432  inches  ? 

37.  How  many  yards  in  7  miles  12  rods  ? 

38.  How  many  acres  in  5  square  miles  ? 

39.  How  many  cubic  feet  in  the  walls  of  a  brick  house  42  feet 
long,  30  feet  wide,  25  feet  high,  the  walls  being  1  foot  thick  ? 

40.  How  many  barrels  in  6481  pints  ? 


REDUCTION.  97 

41.  How  many  pints  in  25  barrels  of  beer  ? 

42.  How  many  quarts  in  15  bushels,  3  pecks  ? 

43.  How  many  days  in  72000  minutes  ? 

44.  How  many  minutes  in  27°  12'  ? 

EXERCISE  rv. 

45.  How  many  shillings  in  260  farthings  ? 

46.  How  many  shillings  in  £12  10s.  ? 

47.  How  many  ounces  in  12  Ibs.  10  oz.  of  gold  ? 

48.  How  many  ounces  in  12  Ibs.  10  oz.  of  tea  ? 

49.  How  many  ounces  in  856  scruples  ? 

50.  How  many  ells  English  in  5  yards  ? 

51.  How  many  inches  in  1  M.  32  rods  ? 

52.  How  many  square  inches  in  5  rods  square  ?  in  5  square 
rods  ? 

53.  How  many  cubic  feet  in  the  walls  of  a  stone  house,  30 
feet  long,  26  feet  wide,  and  24  feet  high,  the  walls  being  1  foot 
thick  ? 

54.  How  many  gallons  in  1  pipe,  1  hhd.  1  barrel  ? 

55.  How  many  barrels  in  648  gallons  of  beer  ? 

56.  How  many  bushels  in  3840  quarts  ? 

57.  How  many  days  in  5  years  ? 

58.  How  many  signs  in  1800'  ? 

EXEECISE  v. 

59.  How  many  pence  in  £10  10s.  ? 

60.  How  many  pence  in  500  farthings  ? 

61.  How  many  pounds  in  5672  pwt.  ? 

62.  How  many  pounds  in  4  T.  16  cwt.  21  Ibs,  ? 

63.  How  many  pounds  in  6742  gr.  of  camphor  ? 

64.  How  many  yards  in  672  nails  ? 

65.  How  many  rods  in  3>^  miles  ? 

66.  How  many  rods  in  10,672  inch  ? 

67.  How  many  acres  in  a  field  containing  320  sq.  rods  ? 

68.  How  many  acres  in  a  field  160  rods  sq.  ? 

69.  How  many  cubic  feet  in  a  ditch  around  a  garden  6  rods 
long  and  4  rods  wide,  the  ditch  being  2  feet  wide  and  3  feet 
deep? 

5 


98  REDUCTION. 

70.  How  many  gallons  in  4728  gills  ? 

71.  How  many  gallons  in  2  hhd.  1  barrel  of  beer  ? 

72.  How  many  pints  in  15  bushels  6  quarts  ? 

73.  How  many  months  in  273456  minutes  ? 

74.  How  many  seconds  in  27°  30'  ? 

EXEECISE  VI. 

75.  How  many  farthings  in  £18  7s.  10}£d.  ? 

76.  How  many  grains  in  11  ounces  of  gold  ? 

77.  How  many  hundredweight  in  12760  ounces  of  flour  ? 

78.  How  many  grains  in  56  ounces  of  morphine  ? 

79.  How  many  ells  English  in  10  ells  Flemish  ? 

80.  How  many  feet  in  312  inches  ? 

81.  How  many  feet  in  275  rods  ? 

82.  How  many  square  inches  of  gilding  will  cover  the  frame 
of  a  looking  glass  36  inches  long,  20  in.  wide,  the  width  of  the 
frame  being  4  inches  ? 

83.  How  many  cords  of  wood  in  pile  64  feet  long,  4  feet  wide 
and  5  feet  high  ? 

84.  How  many  quarts  in  9  hhds.  3  qts.  ? 

85.  How  many  barrels  in  3472  qts.  of  beer  ? 

86.  How  many  pecks  712  pints  ? 

87.  How  many  seconds  in  1  month  ? 

88.  How  many  degrees  in  56312  seconds  of  a  circle  ? 

EXEECISE  vn. 

89.  How  many  shillings  in  187^  pence  ? 

90.  How  many  ounces  in  6512  grains  of  silver  ? 

91.  How  many  ounces  in  11  cwt.  12  Ibs.  ? 

92.  How  many  ounces  in  678  grains  of  laudanum  ? 

93.  How  many  nails  in  1%  yards  ? 

94.  How  many  rods  in  12^  miles  ? 

95.  How  many  rods  in  1428  feet  ? 

96.  How  many  acres  in  a  town  5  miles  square  ? 

97.  How  many  acres  in  6  square  miles  ? 

98.  How  many  solid  feet  in  72  tons  of  hewn  timber  ? 

99.  How  many  bogheads  in  6854.gallons  ? 

100.  How  many  half -pints  in  2  barrels  of  beer  ? 

101.  How  many  pints  in  36  bushels  ? 


PROMISCUOUS  EXAMPLES.  99 


102.  How  many  hours  in  36000  seconds  ? 

103.  How  many  minutes  in  *>£  a  circle  ? 


Promiscuous  Examples  in  Reduction  of  Compound 
Numbers  occurring  in  business,  &c, 

EXERCISE  Vm. 

104.  A  silversmith  made  a  gold  cup  weighing  8  oz,  10  pwt. ; 
what  did  it  cost  at  5  cts.  a  grain  ? 

105.  A  manufacturer  bought  7  cwt.  16  Ibs.  of  wool ;  what 
did  it  cost  at  37 1.<  cts.  a  pound  ? 

106.  A  druggist  sold  5  dr.  2  sc.  morphine  ;  what  was  the 
amount  at  8  mills  a  grain  ? 

107.  If  a  tailor  uses  4  yds.  2  qrs.  of  cloth  in  making  an  over- 
coat, how  many  can  he  make  from  29  yards  ? 

108.  A  contractor  agreed  to  make  a  road  for  $325.50  a  mile  ; 
what  was  the  price  per  rod  ? 

109.  A  farmer  sold  3  acres  1  K.  of  land  for  45  cts.  a  square 
rod  ;  what  did  he  receive  for  it  ? 

110.  At  $5  a  cord,  what  costs  1  cord  foot  of  wood  ? 

111.  A  merchant  bought  15  gallons  2  quarts  of  oil  at  1  shil- 
ling a  pint ;  what  did  it  all  cost  ? 

112.  If  3  bushels  4  quarts  of  salt  cost  $4,  what  is  the  price 
per  pint  ? 

113.  If  a  man's  income  is  $1200  a  year,  how  much  is  it  per 
day? 

114.  If  a  star  move  from  W.  to  E.  at  the  rate  of  10'  30"  daily, 
how  long  will  it  be  in  completing  a  circle  ? 

115.  A  stationer  sells  paper  for  20  cents  a  quire  ;  what  will  3 
reams  cost  ? 

EXEECISE   IX. 

116.  A  silversmith  paid  $300  for  1  Ib.  of  gold  ;  what  was  the 
price  per  grain  ? 

117.  A  blacksmith  used  8  ounces  of  iron  in  making  a  horse 
shoe  ;  how  many  did  he  make  from  60  pounds  ? 


100  REDUCTION. 

118.  Paid  a  druggist  $1.25  for  4  ounces  of  jalap ;  what  was 
the  price  per  grain  ? 

119.  A  merchant  tailor  made  12  vests,  each  containing  3  qrs. 
of  silk  velvet ;  how  many  yards  did  he  use  ? 

120.  A  telegraphic  company  paid  $1250  for  wire  at  3  cents  a 
yard;  how  many  miles  did  it  extend  ? 

121.  A  man  bought  a  field  150  rods  long  and  75  rods  wide, 
at  $64  an  acre  ;  what  did  it  cost  ? 

122.  A  ditcher  agreed  to  dig  a  ditch  60  rods  long,  2  ft.  wide, 
and  3  ft.  deep,  for  $44  ;  what  did  he  receive  for  each  solid  yard  ? 

123.  A  brewer  sold  5  hogsheads,  15  gallons  of  ale  at  4  pence 
a  quart ;  what  did  he  receive  for  it  ? 

124.  A  woman  sold  a  quantity  of  blackberries  for  $6.25,  at 
6  cents  a  quart ;  how  many  bushels  did  she  sell  ? 

125.  A  laborer  earned  $56  in  chopping  wood  at  75  cents  a 
day;  how  many  months  did  he  work  ? 

126.  The  apparent  motion  of  the  sun  being  one  circle  in  the 
heavens  a  year,  how  many  seconds  does  it  move  daily  ? 

127.  What  will  6  gross  of  buttons  cost  at  8  pence  a  dozen  ? 

EXEECISE  x. 

128.  If  a  silversmith  uses  8  ounces  of  silver  in  making  a  cup, 
how  many  cups  can  he  make  from  30  pounds  ? 

129.  If  a  family  consumes  2  pounds  12  ounces  of  flour  daily, 
how  long  will  1  barrel  Kst  them  ? 

130.  How  many  powders,  each  composed  of  4  grains,  can  be 
made  from  a  mixture  of  medicine  weighing  1  Ib.  6  oz.  2  dr.  1  sc.  ? 

131.  How  many  vests,  each  containing  3  quarters,  can  be 
made  from  a  piece  of  valentia  measuring  16>£  yards  ? 

132.  At  12)^  cents  a  foot,  how  much  will  a  lead  pipe  cost 
extending  from  a  house  to  a  spring  ^  mile  distant  ? 

133.  How  many  panes  of  glass,  8  inches  by  10  inches,  are  in 
a  box  containing  100  square  feet  ? 

134.  What  is  a  pile  of  wood  worth  which  is  112  feet  long,  4 
feet  wide,  and  6  feet  high,  at  $4^  a  cord  ? 

135.  How  many  bottles,  containing  3  pints,  each  can  be  filled, 
from  a  hogshead  of  wine  ? 


PROMISCUOUS   EXAMPLES.  101 

136.  At  6  cts.  a  quart,  how  much  will  51^  bushels  of  salt  cost  ? 

137.  If  a  carriage  wheel  turn  round  once  in  passing  over  8 
feet  9  inches,  how  many  times  will  it  turn  in  going  3%  miles  ? 

138.  If  a  clock  tick  4  times  in  a  second,  how  many  times 
does  it  tick  in  a  day  ? 

139.  A  stationer  sells  paper  for  $4  a  ream  ;  how  much  is  it  a 
quire  ? 

EXEECISE   XI. 

140.  At  4d.  a  grain,  how  many  pounds  sterling  will  a  gold 
cup  cost  weighing  7  oz.  10  pwt.  ? 

141.  If  a  pound  of   wool  cost  4s.  6d.,  how  many  pounds 
sterling  will  8  cwt.  3  qrs.  cost  ? 

142.  At  >£d.  a  grain,  how  much  will  2  oz.  5  dr.  2  sc.  of  qui- 
nine cost '? 

143.  At  lOd.  a  foot,  how  much  will  a  gas  pipe  cost  measur- 
ing 3  rds.  4yds.? 

144.  At  10s.  6>£d.  a  square  yard  of  oil  cloth,  how  much  will 
it  cost  to  cover  a  hall  floor  30  feet  by  6  feet  ? 

145.  At  8d.  a  solid  foot,  how  much  will  a  ton  of  hewn  tim- 
ber cost  ? 

146.  If  a  pint  of  Port  wine  cost  7s.  6d.,  how  much  will  10)£ 
gallons  cost  ? 

147.  If  a  laborer  earn  6d.  an  hour,  how  much  will  he  earn  in 
a  month  of  working  time  (6  days  a  week  and  10  hours  a  day  ?) 

148.  If  10  gross  of  buttons  cost  $6,  what  is  the  price  per  doz.  ? 

EXERCISE  XII. 

149.  If  a  silversmith  buy  1  pound  of  silver  for  $40,  and  sell 
it  for  1  cent  a  grain,  how  much  will  he  gain  ? 

150.  If  a  merchant  buy  10  cwt.  2  qrs.  of  sugar  for  $115.50, 
for  what  must  he  sell  it  to  gain  1  cent,  a  pound. 

151.  If  a  druggist  pay  8  mills  a  grain  for  calomel  and  sell  it 
for  9  mills,  how  much  will  he  gain  on  a  pound  ? 

152.  A  contractor  agreed  to  make  a  road  7^  miles  long  for 
$2500  and  paid  his  laborers  $1  a  rod  ;  how  much  did  he  gain  ? 

153.  A  speculator  bought  city  lots  containing  lj^  acres  for 


102  BEDUCTION. 

$50,000,  and  sold  them  for  $1  a  square  foot ;  how  much  did  he 
gain? 

154.  A  job  mason  agreed  to  build  the  foundation  of  a  house 
32  feet  long,  24  feet  wide,  the  foundation  to  be  9  feet  high,  18 
inches  thick,  for  $12,  but  was  obliged  to  pay  his  journeymen 
1  cent  a  solid  foot ;  how  much  did  he  lose  ? 

155.  A  merchant  bought  3  hhds.  of  molasses  for  £28  7s.  and 
sold  it  for  1  shilling  a  quart,  how  much  did  he  gain  ? 

156.  A  merchant  in  selling  peas  at  20  cents  a  quart  gains  5 
cents;  what  did  they  cost  a  bushel  ? 

157.  If  a  watch  lose  2  seconds  an  hour,  how  much  will  it 
lose  in  a  week  ? 

EXEKCISE   XTTT. 

158.  A  silversmith  paid  $175.68  for  gold  ore  at  three  cents  a 
grain  ;  how  much  did  he  buy  ? 

159.  A  merchant  bought  1  T.  1  cwt.  of  rice  at  9d.  a  pound  ; 
what  did  it  all  cost  ? 

160.  A  druggist  sells  opium  for  4d.  a  scruple  ;  how  many 
ounces  can  be  bought  for  £2  10s.  ? 

161.  How  many  suits  of  clothes,  each  containing  5  yards  3 
quarters  ;  can  be  cut  from  69  yards  ? 

162.  How  much  will  it  cost  to  make  a  wire  fence  4  wires 
high,  around  a  field  40  rods  long  and  32  rods  wide  at  3  cents 
a  yard  ? 

163.  What  will  an  acre  of  ground  cost  at  75  cents  a  square 
yard  ? 

164.  How  many  cubic  inches  in  a  block  of  marble  1  yard 
long,  2  feet  wide,  18  inches  thick  ? 

165.  What  will  a  hogshead  of  vinegar  cost  at  5  mills  a  gill  ? 

166.  What  will  2  bushels  3  pecks  of  plums  cost  at  4  cent* 
a  pint  ? 

167.  If  a  ship  sail  12  miles  an  hour,  how  far  will  she  sail 
in  2  months,  1  week,  3  days. 

168.  If  a  planet  move  1°  30'  a  day ;  how  long  will  it  be  in 
moving  through  each  sign  of  the  Zodiac  ? 

169.  If  one  gross  of  steel  pens  cost  72d. ,  what  does  1  pen  cost  ? 


PROMISCUOUS   EXAMPLES.  103 

EXERCISE   XIV. 

170.  If  24  English  Bibles  cost  22  guineas,  6s.,  what  is  the 
price  of  each  ? 

171.  A  silver  dollar  contains  412^  grains  ;  how  many  dollars 
can  be  made  from  25  Ibs.  15  pwt.  15  gr.  of  silver  ? 

172.  In  6  tons  14  cwt.  3  qrs.  9  Ibs.  of  butter,  how  many 
firkins  ? 

173 .  How  many  acres  in  a  field  430  rods  long  and  132  feet  wide  ? 

174.  How  many  yards  of  muslin  3  quarters  wide  will  line  10 
yards  of  merino  1|  yards  wide  ? 

175.  How  many  sheets  of  tin  15  inches  by  12  inches,  will  cover 
the  roof  of  a  house  40  feet  long  and  25  feet  wide,  the  rafters  on 
each  side  being  16  feet  long  ? 

176.  How  many  blocks  of  granite  3  feet  long,  21  inches  wide, 
and  18  inches  thick,  will  it  take  to  build  the  walls  of  a  church 
80  feet  long,  62  feet  wide,  36  feet  high,  the  walls  being  18 
inches  thick  ? 

177.  If  a  barrel  of  ale  cost  $7.25,  what  will  be  gained  by  re- 
tailing it  a  4  cents  a  pint  ? 

178.  How  long  would  it  take  a  locomotive  to  travel  3000 
miles,  at  the  rate  of  3  rods  a  second  ? 

179.  If  a  ream  of  paper  cost  $4  and  be  sold  for  25  cents  a 
quire,  how  much  would  be  gained  ? 

180.  At  5  cents  a  dozen,  how  many  gross  of  buttons  can  be 
bought  for  $3  ? 

EXEKOISE  XV. 

181.  How  many  yards  of  carpeting  2  feet  wide,  will  cover 
a  room  24  feet  by  18  feet  ? 

182 .  How  many  pieces  of  paper  9  yards  long,  15  inches  wide, 
will  cover  the  sides  of  a  room  18  feet  long  15  feet  wide,  and  11 
feet  high,  there  being  one  door  6  feet  3  inches  by  2  feet  9 
inches,  and  two  windows  5  feet  by  4  feet  ? 

183.  How  many  square  yards  of  plastering  in  the, same 
room  ? 

184.  How  much  will  it  cost  to  pave  a  sidewalk  10  rods  long, 
10  feet  wide,  at  $10  a  square  rod  ? 

185.  If  a  piece  of  land  containing  6  acres  be  divided  into 


104  EEDUCTION. 

building  lots  8  rods  long  and  2  rods  wide,  and  each  be  sold 
for  $300,  for  how  much  will  it  all  be  sold  ? 

186.  How  many  shingles  will  cover  the  roof  of  a  house  30 
feet  long,  the  rafters  on  each  side  being  15  feet  long,  allowing 
one  shingle  for  every  20  square  inches  ? 

187.  How  many  bricks  will  it  require  to  build  a  chimney, 
averaging  4  feet  by  3  feet  and  30  feet  high,  the  walls  being  8 
inches  thick,  and  the  bricks  8  inches  long,  4  inches  wide  and  2 
inches  thick  ? 

EXERCISE  XVI. 

188.  How  many  spoons,  each  weighing  2  ounce  10  pwt.,  can 
be  made  from  5  Ibs.  of  silver  ? 

189.  What  will  6  hogsheads  of  beer  cost  at  3  cents  a  quart  ? 

190.  What  will  6  hogsheads  of  wine  cost  at  1  shilling  a  pint  ? 

191.  How  many  bushels  of  nuts  can  be  bought  for  $20  at  4 
cents  a  quart  ? 

192.  What  will  it  cost  to  plaster  a  room  30  feet  long,  20  feet 
wide  and  8  feet  high,  at  25  cents  a  square  yard,  there  being 
two  doors  7  feet  by  4  feet,  and  two  windows  5  feet  by  3  feet, 
and  a  mop  board  6  inches  wide  ? 

193.  What  is  the  value  of  a  pile  of  wood  108  feet  long,  6  feet 
high,  and  4  feet  wide,  at  $7£  a  cord  ? 

194.  How  many  rods  of  fence  will  inclose  a  tract  of  land  1 
mile  long  and  180  rods  wide  ? 

195.  How  many  acres  in  a  town  5  miles  square  ? 

196.  How  many  acres  in  a  town  6  miles  long  and  3  miles  wide  ? 

197.  How  much  will  it  cost  to  carpet  a  room  24  feet  long, 
18  feet  wide,  the  carpet  being  2  feet  wide  and  the  price  $1£ 
a  yard  ? 

198.  What  will  it  cost  to  dig  a  cellar  40  feet  long  32  feet  wide, 
and  8  feet  deep  at  25  cents  a  cubic  yard  ? 

199.  What  will  it  cost  to  dig  a  ditch  around  a  garden  (out- 
side) which  is  6  rods  long  and  4  rods  wide,  the  ditch  to  be  3 
feet  deep  and  2  feet  wide,  at  -|  cent  a  cubic  foot  ? 

EXERCISE  XVH. 

200.  How  many  yards  of  muslin  3  quarters  wide,  will  it  take 
to  line  21  yards  of  silk  2  quarters  2  nails  wide  ? 


COMPOUND  NUMBERS.  105 

201.  How  much  will  it  cost  to  paper  a  room  27  feet  long,  18 
feet  wide  and  12  feet  high,  each  piece  of  paper  containing  9 
yards,  1  foot  6  inches  wide  and  costing  30  cents  ? 

202.  How  many  bricks  will  pave  a  side-walk  120  rods  long, 
12  feet  wide,  each  brick  being  8  inches  long  and  4  inches  wide  ? 

203.  How  high  must  a  load  of  wood  be  to  contain  a  cord,  if 
it  is  9  feet  long  and  4  feet  wide  ? 

204.  How  wide  must  a  field  be  to  contain  six  acres,  if  it  is 
80  rods  long  ? 

205.  How  many  bricks  in  a  pile  20  feet  long,  12  feet  wide, 
and  10  feet  high,  each  brick  being  8  inches  long  4  inches  wide 
and  2  inches  thick  ? 

206.  How  many  bricks  of  the  same  size  in  a  furnace  chimney 
60  feet  high,  having  four  equal  sides,  averaging  5  feet,  and 
the  walls  being  8  inches  thick  ? 


The  Fundamental  Rules  Applied  to  Compound  Numbers. 

Art.  68. — ADDITION  OF  COMPOUND  NUMBEKS. 
EXAMPLE  1.  What  is  the  sum  of  £9  10s.  7d.  ;  £7  11s.  4d. 
Ifar.  ;  £2  18s.  7£d.  ;  4s.  9d.  3  far. 

£    s.  d.  far. 

Process.  — First,  add  the  farthings,  the  sum  of  9  10  7  0 

which  is  6  farthings  (by  reduction  ascending)  Id.  7  11  4  1 

2  farthings.     Write  the  2  farthings  under  the  2  18  7  2 

column  of  farthings,  and  add  or  carry  the  Id.  to  14  9  3 
the  pence  in  the  next  column.     Add  and  reduce 

(if  possible)  the  other  denominations  in  the  same  Ans.  20  15  4  2 
manner. 

RULE. —  Write  the  numbers  of  the  same  denomination,  or 
name  under  one  another  in  order.  Add  each  denomination, 
beginning  with  the  least,  as  in  simple  numbers,  and  reduce 
the  sum  to  the  next  greater  denomination.  Write  the  re- 
mainder under  the  column  added,  and  carry  the  quotient  to 
the  next  column. 


106  REDUCTION. 

PROOF — As  in  simple  Addition. 

In  writing  compound  numbers,  if  any  denomination  is  wanting, 
supply  its  place  with  a  cipher. 

EXAMPLES. 

2.  Add  5  Ibs.  10  oz.  12  pwt.  9  gr ;  1  Ib.  11  oz.  13  pwt.  7  gr ; 
6  Ibs.  9  oz.  7  pwt.  14  gr.  ;  3  Ibs.  6  oz.  12  pwt.  12  gr. 

3.  Add  9  Ibs.  8  oz.  17  gr.  ;  8  Ibs.  10  pwt.  12  gr.  ;  7  oz.  12  pwt. 
15  gr.  ;  4  Ibs.  5  oz.  21  pwt. 

4.  What  is  the  sum  of  7  T.  19  cwt.  1  qr.  15  Ibs.  ;  10  T.  8  cwt. 

2  qrs.  18  Ib.  ;  12  T.  10  cwt.  3  qrs.  14  Ibs.  ;   5  T.  11  cwt.  1  qr. 

9  Ibs.  ? 

5.  Add  5  cwt.  3  qrs.  16  Ibs.  12  oz.  10  dr.  ;  7  cwt.  1  qr.  18  Ibs. 
14  oz.  ;  9  cwt.  8  Ibs.  10  oz.  10  d.  ;  3  qrs.  18  Ibs.  9  oz.  8  dr. 

6.  Add  3  Ibs.  2  oz.  4  dr.  2  sc.  10  gr.  ;  4  Ibs.  8  oz.  5  dr.  15 
gr.  ;  6  Ibs.  11  oz.  7  dr.  2  sc.  ;  2  oz.  4  dr.  1  sc.  15  gr. 

7.  What  is  the  sum  of  12  yd.  2  qr.  2  na.  ;  8  yd.  3  qr.  1  na. 
4  yd.  1  qr.  2  na.  ;   6  yd.  2  qr.  2  na.  ? 

8.  Add  12  m.  6  fur.  18  rod.  2  ft.  ;  10  m.  4  fur.  16  rod.  10 
ft,  ;  9  m.  7  fur.  15  rod.  9  ft.  ;  7  m.  5  fur.  10  rod.  12  ft. 

9.  How  many  acres  in  four  fi  Ms,  the  first  containing  6  A. 

3  E.  25  rod.  ;  the  second  5  A.  1  E.  30  rds.  ;  the  third  4  A.  2  E. 
35  rds.  ;  the  fourth  7  A.  3  E.  28  rd.  ? 

10.  How  many  cords  in  three  piles  of  wood  measuring  as 
follows.    8  C.  24  ft. ;  7  C.  64  ft.  ;  9  C.  84  ft.  ? 

11.  What  is  the  sum  of  10  hhd.  42  gal.  1  qt.  1  pt.  ;  9  hhd. 
30  gal.  1  pt.  2  giUs  ;  54  gal.  3  qt.  1  pt.  2  gills  ;  6  hhd.  25  gal. 
2  qt.  1  pt.  ? 

12.  Add 5  y.  4  mo.  12  d.  16  m.  20  sec.  ;  4  y .  3  mo.  15  d.  45  m.  ; 
6  y.  6  mo.   20  d.  14  m.  ;  6  m.  6d.  6  h.  40  sec.  ;  7  y.  28  d.  20  m. 

10  sec. 

13.  What  is  the  difference  between  45°  16'  28"  north  latitude 
and  30°  43'  32"  south  latitude  ? 

Art.  69, — SUBTRACTION  OF  COMPOUND  NUMBERS. 
EXAMPLE  1.  From  3  hhds.  18  gals.  1  qt.  1  pt.  1  gi.,  take  1 
hhd.  31  gals.  2  qts.  1  pt.  3  gi. 


SUBTRACTION  OF  COMPOUND  NUMBERS.      107 

Process.— Since  3  gills  cannot  be  taken  from  hhd.gal.qt.pt.gi. 
1  gill,  add  1  pint  reduced  to  4  gills,  to  the  1  3  18  1  1  1 

gill,  then  subtract  the  3  gills,  and  write  the  re-  1    31    2    1    3 

mainder,   2  gills,  underneath,  also  carry  the  1  £m  1    49    2    1    2 
pint  to  the  next  denomination.     Thus  proceed. 

RULE. —  Write  the  less  compound  number  under  the  greater, 
so  that  numbers  of  the  same  name  will  be  under  one  another. 
Subtract  each  denomination  as  in  simple  numbers.  If  any 
number  is  greater  than  the  one  above  it,  add  to  the  upper 
number  1  of  the  next  denomination  reduced  to  the  same  name, 
then  subtract  and  carry  1  to  the  next  denomination. 

PKOOF — .4s  in  simple  Subtraction. 

EXAMPLES. 

2.  From  £11  12s.  8d.  2  far.,  subtract  £4  15s.  6d.  3far. 

3.  From  161bs.  10  oz.  14pwts.  20grs.,  subtract  81bs.  8  oz. 
20  pwts.  10  grs. 

4.  What  is  the  difference  between  17  T.  11  cwt.  1  qr.  10  Iba. 
and  10  T.  15  cwt.  3  qr.  6  Ibs.  ? 

5.  What  is  the  difference  between  16  Ibs.  8  oz.  6  ^dr.  2  sc. 
and  8  Ibs.  4  oz.  6  dr.  12  grs.  ? 

6.  From  14  yds.  1  qr.  2  ua.  subtract  8  yds.  2  qrs.  3  na. 

7.  From  127  A.  1  B.  30  rds.  subtract  56  A.  2  E.  24  rds. 

8.  What  is  the  difference  between  4  y.  5  mo.  10  d.  6  h.  and 
1  y.  9  mo.  21  d.  45  m.  ? 

9.  What  is  the  difference  between  203 12'  24"  and  5°  15'  45'? 

Art»  70. — To  find  the  time  between  two  dates. 

Ex.  10.  How  old  was  a  person,  born  May  15,  1820,  when  he 
died,  Oct.  21,  1856  ? 

Process.— (October  is  the  10th  and  May    Died  1856  10  21 
the  5th  month.  1  Born  1820    5  15 

Age     ~~36    5     6  Ans. 

EULE. — Subtract  the  first  date  from  the  last,  numbering  the 
months  in  their  order. 
30  days  is  considered  a  month,  and  12  months  a  year. 


108  REDUCTION. 

EXAMPLES. 

11.  A  note  dated  Jan.  1,  1860,  was  due  July  15,  1861 ;  for 
what  time  was  it  given  ? 

12.  A  father  was  born  May  16,  1815,  and  his  son  June  23, 
1855  ;  how  much  older  is  the  father  than  his  son  ? 

13.  A  note  dated  Aug.  15,  1865  was  due  May  1,  1866 ;  for 
what  time  was  it  given  ? 

14.  A  man  was  born  April  12,  1828,  and  his  wife  Sept.  21, 
1833  ;  how  much  older  is  the  man  than  his  wife  ? 

15.  On  the  19th  of  June,  1860,  a  man  gave  his  note  to  be 
paid  July  1,  1801 ;  how  long  was  the  time  ? 

Art.  71. — MULTIPLICATION  or  COMPOUND  NUMBEBS. 

EXAMPLE  1. — Bought  5  barrels  of  molasses,  each  containing 
34  gal.  3  qt.  1  pt.  2  gills  ;  how  much  did  they  all  contain  ? 

Process. — The  same  as  in  addition,  except  hhd.gal.qt.pt.gi. 

we  multiply  instead  of  adding  each  denomi-  0  34  3  1  2 

nation. 

Ans.  2    46    2    1~2 

RULE. — Multiply  each  denomination  separately,  as  in  sim- 
ple multiplication,  beginning  with  the  least,  and  reduce  the  pro- 
duct to  the  next  greater  denomination.  Write  the  remainder 
under  the  number  multiplied,  and  carry  the  quotient  to  the 
next  denomination. 

PROOF. —  The  same  as  in  simple  Multiplication. 
The  multiplier  is  properly  an  abstract  number,  though  it  may  be 
of  a  particular  denomination  or  name. 

EXAMPLES. 

2.  Multiply  3  oz.  12  pwts.  18  grs.  by  5. 

3.  Multiply  14  cwt.  1  qr.  15  Ibs.  by  6. 

4.  Multiply  4oz.  3  drs.  2  sc.  10  grs.  by  7. 

5.  Multiply  3  yds.  1  qr.  2  na.  by  8. 

6.  Multiply  2  m.  3  fur.  9  rods,  8  ft.  by  9. 

7.  Multiply  26  A.  2  B.  20  rods  by  1*0. 

8.  Multiply  5  0.  112  ft.  by  11. 

9.  Multiply  7  hhds.  45  gals.  2  qts.  by  12. 

10.  8  y,  6  mo.  7  d.  30  min.  by  15. 

11.  12°  12'  12"  by  8. 


DIVISION   OF  COMPOUND   NUMBERS. 


109 


Art.  72. — DIVISION  OF  COMPOUND  NUMBERS. 

CASE  1.  To  divide  a  compound  number  into  any  number 
of  equal  parts. 

EXAMPLE  1. — Divide  12  cwt.  2  qr.  18  Ibs.  equally  among  5 
persons. 

Process.— 12  cwt. -^-5=2  cwt.  and  2  cwt.  5)12  cwt.  2  qrs.  18  Ibs. 
remainder;  write  the  quotient  under  cwt.  2  "  2  "  3f  " 

and  reduce  the  remainder  to  quarters  ;  2 

cwt;X4=8  qrs.,  to  which  add  the  2  qrs.  and  the  sum  will  be  10  qrs., 
which  divide  and  thus  proceed. 

RULE. — Divide  each  denomination  separately,  beginning 
with  the  greatest,  and  write  the  quotient  under  it.  Reduce  the 
remainder  to  the  next  less  denomination,  to  which  add  it  and 
divide  again. 

PROOF. — The  same  as  in  Simple  Division. 
Ex.  Divide  48  T.  10  cwt.  by  25. 


Process. — The  same  except  by  long  division. 


T.     cwts. 
25)48    10(1  T. 
25 

23 

20 


25)470(18  cwt. 
25 


Ans.  1  T.  18  cwt.  3  qr.  5  Ib. 

EXAMPLES. 

3.  Divide  £100  15s.  lOd.  by  4. 

4.  Divide  10  Ibs.  8  oz.  10  pwt.  by  6. 

5.  Divide  12  cwt.  1  qr.  14  Ibs.  by  9. 


200 
~W 
4 

25)80(3  qrs. 
'75 
5 
25 

25)125(5  Ibs. 
125 


110  REDUCTION. 

6.  Divide  4  Ibs.  6  oz.  4  dr.  2  scr.  by  11. 

7.  Divide  15  yds.  2  qrs.  3  na.  by  15, 

8.  Divide  16  m.  3  fur.  20  rd.  12  ft.  by  21. 

9.  Divide  124  A.  3  B.  150 rd.  by  100. 

10.  Divide  50  C.  72  ft.  by  16. 

11.  Divide  1200  bu.  3  pk.  by  175. 

12.  Divide  9  hhd.  by  11. 

CASE  2.   To  divide  one  compound  number  by  another. 
Ex.  13.  Divide  £7  10s.  among  as  many  persons  as  possible, 
giving  to  each  12s.  6d. 

Process. — Reduce  both  the  dividend  and  divisor  to  pence,  and  then 
divide. 

Since  this  is  performed  by  reduction  and  simple  division  it  is  not 
usually  included  in  Conpound  Division. 


Art,  73,— Special  Application  of  Compound  Numbers  to 
Longitude  and  Time. 

Since  the  apparent  motion  of  the  sun  around  the  earth 
is  in  a  circle  of  360°  in  24  hours — 

360°-f  24  or  15°=1    hour  of  time. 

1°=JL  hour  or  4  minutes  of  time. 
15°_:_60  or  15'  =1    minute  of  time. 

1'  — _L  minute  or  4  seconds  of  time. 
15' -H>0  or  15 "—I    second  of  time. 

Art,  74, — To   FIND    THE   DIFFERENCE   OF   TIME  WHEN  THE 
DIFFERENCE  OF  LONGITUDE  is  KNOWN. 

EXAMPLE  1.  The  difference  of  longitude  between  two  places 
is  18°  35' ;  what  is  the  difference  of  time  ? 

Process.- 1st,  since  15C=1  hour,  18°=1  hour    15)18°  35'      0" 
and  3°  remainder;  3°=180,  to  which  add  the  35'          ib.  14  m>  20  sec. 
and  the  sum  will  be  215' ;  and  since  15'=1  min. 
215'=14  niin.  and  5'  remainder,  which  reduced  to  (")  and  divided, 
is  equal  to  20  sec. 

Process.— 2d,  since  l'=4  sec.  35'=140  sec.  or  2  18°    35' 

min.  20  sec.    And  since  1°:=4  min.  18°=72  min.  to  _ 
which  add  the  2  min.  and  the  sum  will  be  74  min.  =  j  jj  14  m  20  sec 
1  h.  14  min.    Therefore  18°  35'=1  h.  14  min.  20  sec. 


SPECIAL  APPLICATION  OF  COMPOUND  NUMBERS.     Ill 

RULE. — Divide  the  longitude  by  15  as  in  Compound  Num- 
bers and  the  quotient  of  degrees  will  be  hours,  of  minutes  (' ) 
minutes  of  time,  and  of  seconds  (" )  seconds  of  time. 

Or  Multiply  the  longitude  by  4,  and  the  product  of  min- 
utes (' )  will  be  seconds  of  time,  and  of  degrees  minutes  of 
time,  which  may  be  reduced  to  hours. 

EXAMPLES. 

2.  The  difference  of  longitude  between  Boston  and  London 
is  71°  4';  what  is  the  difference  of  time  ? 

3.  The  difference  of    longitude  between  Washington  and 
London  is  76°  53 ;  what  is  the  difference  of  time  ? 

4.  The  difference  of  lou  ntude  between  New  York  and  San 
Francisco  is  51°  26';  what  is  difference  of  time  ? 

When  the  longitudes  of  two  places  are  given,  their  difference  may  be 
found  by  subtraction  when  they  are  both  East  or  West ;  by  addition 
when  one  is  East  and  the  other  West. 

Art,  75, — To   FIND   THE   DIFFERENCE  OF  LONGITUDE  WHEN 
THE  DIFFERENCE  OF  TIME  is  GIVEN. 

EXAMPLE  5.  The  difference  of  time  between  Philadelphia 
and  Cincinnati  is  37  min.  20" ;  what  is  the  difference  of  longi- 
tude? 

m.  sec. 

Process.— 1st,  since  1  sec.  =15"  of  long.  20  sec.=  37  20 

300"  =5'  and  since  1  min.  =15'  of  long.  37  min.  =555  15 

min.,  to  which  add  the  5'  and  the  sum  will  be  560'=     £ns  9°  2(F~"~ 
9°  20' 

m.  sec. 

Process.—  2d,  since  4  min.=l°  of  long.   37  min.          4)37    20 
=9°  and  1  min.  rem.,  1  min.  =60  sec.  and  20  sec.     A^      9°~2(F~ 
more  are  80  sec.,  and  since  4  sec.=l'  of  long.   80 
sec.  =20'  of  long.     Therefore  37  min.  20  sec.  =9°  20'  of  long. 

RULE. — Multiply  the  time  by  15,  as  in  Compound  Numbers, 
and  the  product  of  seconds  of  time  will  be  ( " )  seconds  of  longi- 
tude, of  minutes  (')  minutes  of  longitude,  and  of  hours,  of 
degrees  of  longitude. 

Or  Multiply  the  hours  by  15  to  find  degrees,  and  divide 
the  minutes  by  4  to  find  more  degrees,  and  the  seconds  by  4  to 
find  minutes  of  longitude. 


112  PROMISCUOUS   EXAMPLES. 

EXAMPLES. 

6.  The  difference  of  time  between  Washington  and  Cincin- 
nati is  29  minutes  36  seconds  ;  what  is  the  difference  of  long- 
tude  ? 

7.  The  difference  of  time  between  Washington  and  London 
is  5  hours,  8  minutes,  4  seconds;  what  is  the  difference  of 
longitude  ? 

8.  The  difference    of    time  between    New  York  and  San 
Francisco  is  3  hours,  25  minutes,  44  seconds ;  what  is  the 
difference  of  longitude  ? 


Art,  76,— Promiscuous  Examples  in  Addition,  Sub- 
traction, Multiplication,  and  Division  of  Compound 
Numbers. 

EXEBCISE  I. 

1.  A  man  when  married  was  20  years,  9  months,  12  days 
old ;  he  and  his  wife  lived  together  21  years,  4  months,  15 
days,  and  he  lived  after  the  death  of  his  wife  12  years,  8  months, 

3  days  ;  what  was  his  age  when  he  died  ? 

2.  A  railroad  when  completed  is  to  be  224  miles  long.      The 
part  already  finished  is  109  miles,  3  furlongs  18  rods;  how  much 
remains  to  be  made  ? 

3.  A  druggist  sold  9  boxes  of  medicine,  each  weighing  3 
ounces,  4  drams,  2  scruples ;  what  was  the  whole  weight  ? 

4.  A  wholesale  grocer  shipped  1  ton  of  sugar  in  9  barrels  of 
equal  size  ;  how  much  was  there  in  each  barrel  ? 

5.  A  note  dated  July  10,   1860,  was  due  January  1,  1862  ; 
how  long  was  the  time  ? 

6.  The  longitude  of  Jerusalem  is  35°  32   East,  and  that  of 
Baltimore  76°  37'  West,  what  is  the  difference  of  time  ? 

7.  The  difference  of  time  between  Boston  and  London  is 

4  hours  44  minutes  32  seconds,  what  is  the  difference  of  longi- 
tude ? 

EXEECISE  n. 

8.  Bought  4  pieces  of  cloth  measuring  as  follows  23  yards, 


COMPOUND   NUMBERS.  113 

3  quarters,  2  nails  ;  27  yards,  2  quarters,  1  nail ;  25  yards,  20 
quarters,  3  nails;  how  many  yards  in  all  of  them  ? 

9.  A  party  started  on  a  journey  of  315  miles.    After  travelling 
156  miles,  4  furlongs,  20  rods,  how  much  farther  had  they  to 
travel  ? 

10.  A  farm  is  divided  into  16  fields,  and  each  field  contains 
9  acres,  130  square  rods  ;  how  large  is  the  farm  ? 

11.  A  man  wishes  to  draw  12  cords  of  wood  in  14  loads;  how 
much  must  he  draw  in  each  load  ? 

12.  A  note  was  dated  June  15,  1861,  and  was  paid  June  1st, 
1862  ;  how  long  was  the  time  ? 

13.  The  difference  of  longitude  between  Washington  and 
London  is  77°,  what  is  the  difference  of  time  ? 

14.  The  difference  of  time  between  New  York  and  San  Fran- 
cisco is  3  hours,  25  minutes,  44  seconds  ;  what  is  the  difference 
of  longitude  ? 

EXERCISE  HE. 

15.  A  farmer  carried  to  market  four  loads  of  corn;  in  the 
first  36  bushels,  2  pecks,  4  quarts  ;  in  the  second  42  bushels,  3 
pecks,  1  quart ;  in  the  third  38  bushels,  1  peck,  3  quarts ; 
in  the  fourth  40  bushels,  6  quarts  ;  how  many  bushels  in  all  ? 

16.  A  merchant  has  sold  from  a  hogshead  of  molasses  38 
gallons,  1  quart,  1  pint ;  how  much  is  unsold  ? 

17.  What  is  the  weight  of  a  dozen  silver  spoons,  each  weigh- 
ing 6  ounces,  8  pennyweights,  10  grains  ? 

18.  If  a  pound  of  rhubarb  be  divided  into  100  doses  ;  how 
much  will  a  dose  contain  ? 

19.  John  Q.  Adams,  was  born  July  11,  1767  ;  what  was  his 
age  when  he  died,  February  23,  1848  ? 

20.  What  is  the  difference  in  the  time  of  two  places,  one  in 
longitude  56°  30',  and  the  other  in  longitude  125°  20',  both 
East? 

21.  When  it  is  noon  at  Greenwich  England,  a  captain  of  a 
ship  finds  that  is  4  o'clock  P.M.  where  he  is ;  in  what  longitude 
is  he  ? 

EXERCISE   TV. 

22.  Two  men  building  a  stone  fence,  built  the  first  day  5 


114  PROMISCUOUS  EXAMPLES. 

rods,  8  feet,  9  inches  ;  the  second  6  rods,  4  feet,  6  inches  ;  the 
third  6  rods  ;  and  the  fourth  they  finished  it,  building  4 
rods  8  inches ;  how  long  was  the  fence  ? 

23.  A  merchant  tailor  bought  a  piece  of  cloth  supposed  to 
contain  27  yards,  2  quarters,  but  on  measuring  it  he  found  that 
it  contained  only  25  yards,  3  quarters ;  how  much  did  it  fall 
short  ? 

24.  A  lumber  merchant  sold  10  loads  of  hewn  timber,  each 
containing  1  ton,  32  feet,  1600  inches  ;  how  much  did  he  sell  ? 

25.  A  farmer  having  256  acres  of  land,  divided  it  equally 
among  his  seven  children  ;  how  much  did  he  give  each  ? 

26.  A  note  was  dated  May  16,  1858,  and  was  due  October 
1st,  1860 ;  how  long  was  the  time  ? 

27.  What  is  the  difference  of  time  in  two  places,  one  36° 
30  East  longitude,  and  the  other  40°  15'  West  longitude  ? 

28.  A  sea  captain  found  that   it  was  20  minutes    past  9 
o'clock  where    he  was,   when    it  was    noon    at  Greenwich, 
England.     In  what  direction  from  Greenwich  was  he,  and  in 
what  longitude  ? 


29.  Bought  four  loads  of  coal,  weighing  as  follows  :  18  cwt.  3 
qrs.  20  Ibs.  ;  16  cwt.  1  qr.  12  Ibs.  ;  19  cwt.  21  Ibs.  ;  17  cwt.  2 
qrs.  9  Ibs.  ;  18  cwt.  3  qrs.  ;  how  much  in  all  ? 

30.  A  merchant  bought  12  hogsheads,  36  gallons  of  oil,  and 
sold  7  hogsheads,  42  gallons,  3  quarts,  how  many  had  he  left  ? 

31.  At  £1  12s.  8d.  a  yard,  how  much  will  20  yards  of  cloth 
cost  ? 

32.  If  18  yards  of  cloth  cost  £28,  what  is  the  price  per  yard  ? 

33.  Washington  was  born  February  22,  1732  ;  what  was  his 
age  when  he  died,  December  14,  1799  ? 

34.  The  longitude  of  Boston  is  71°  4',  and  that  of  St.  Louis 
is  90°  15,  what  is  the  difference  of  time  ? 

35.  A  sea  captain  having  sailed  from  Philadelphia,  longitude 
75°  10 ,  finds  his  watch  1  hour  20  minutes  slower  than  the  time 
where  he  is;  supposing  his  watch  has  kept  good  time,  in  what 
longitude  is  he  ? 


CANCELLATION.  115 


CANCELLATION, 

Art.  77.  —  Cancellation  is  the  omission  of  equal  factors 
in  the  dividend  and  divisor,  or  any  corresponding  terms, 
for  the  purpose  of  shortening  the  operation.  It  is  on  the 
principle  that  dividing  both  the  dividend  and  divisor  by  the 
same  number,  does  not  alter  the  quotient.  (Art.  20,  3.  ) 

EXAMPLE  1.  —  Multiply  12  by  4,  and  divide  the  product  by  4. 


Process.—  12X4=48  and  48-^4=12.      We  therefore 
omit  both  the  multiplication  and  division,  or  cancel  the  —     —  =12 
4,  by  drawing  a  line  across  it,  in  both  the  dividend  and       ** 
divisor. 

Ex.  2.  —  Bought  12  oranges  at  4  cents  each,  and  gave  in  ex- 
change for  them  8  quarts  of  nuts  ;  what  was  the  price  of  the 
nuts  per  quart  ? 

Process.  —  12X4=48,  the  price  of  all  the  oranges,  and  6 

48-f-8=6  the  price  of  the  nuts  per  quart,  but  we  cancel  X-ttyJL 

the  factor  4  by  drawing  a  line  across  it  in  the  dividend  —  -:  —  =6 
and  dividing  the  divisor  8  by  it,  which  gives  the  same 

result  ;  we  also  cancel  the  2.  % 

KULE.  —  Write  the  factors  of  the  divisor  under  those  of  the 
dividend,  and  omit  equal  factors  in  both  the  dividend  and  di- 
visor, or  corresponding  terms,  drawing  a  line  across  them, 
then  proceed  as  in  other  similar  operations. 

Since  cancelling  is  the  same  as  dividing,  the  place  of  a  cancelled 
factor  must  be  supplied  by  1  if  there  is  no  other  factor  left. 

EXAMPLES. 

Cancel  equal  factors  iu  — 

(3.)  2X3X4X6X7  (5.)  4X9XHX16 

8X4X5    X  2  3X  22  X    5 

(4)  4X6X8X10  (6.)  2X13X28 

2X3X4X  2  26X14 

7.  Multiply  together  the  factors  8,  12,  15  and  18,  and  divide 
the  product  by-  the  factors  4,  5  and  9. 


116  CANCELLATION. 

8.  Multiply  the  factors  6,  9,  11  and  12,  and  divide  the  pro- 
duct by  the  factors  3,  4,  3,  4. 

9.  Multiply  the  factors  9,  14,  15,  16,  and  divide  the  product 
by  the  factors  2,  3,  7,  5,  8,  4. 

10.  Multiply  the  factors  11,  24,  22,  30,  and  divide  the  pro- 
duct by  the  factors  4,  5,  44,  33. 

11.  Bought  25  firkins  of  butter,  each  weighing  54  Ibs. ,  at 
30  cts.  a  pound,  and  paid  for  them  in  exchange,  potatoes  in 
barrels,  3  bushels  in  each,  at  75  cts.  a  bushel ;  how  many  bar- 
rels ? 

12.  How  many  boxes  of  starch,  each  containing  20  Ibs.,  at 
12  cts.  a  pound,  will  pay  for  10  sacks  of  corn,  each  containing 
3  bushels,  at  72  cts.  a  bushel  ? 

13.  How  many  bags  of  coffee,  each  weighing  60  Ibs. ,  at  30  cts. 
a  pound,  can  be  bought  for  20  firkins  of  butter,  each  contain- 
ing 90  Ibs.,  at  25  cts.  a  pound  ? 

14.  How  many  barrels  of  sugar,  each  weighing  225  pounds, 
at  16  cts.  a  pound,  can  be  bought  for  25  barrels  of  flour  at  8 
dollars  a  barrel  ? 


Art,  78.— Properties  of  Numbers. 

A  number  is  either  odd  or  even ;  odd  when  it  cannot  be 
exactly  divided  by  2,  and  even  when  it  can  be ,  as  1,  3, 
5,  13,  &c.,  are  odd  numbers ;  2,  4,  8,  12,  &c.,  are  even 
numbers. 

Numbers  are,  also,  either  prime  or  composite. 

A.  prime  number  is  one  which  is  not  the  product  of  any 
numbers  greater  than  1,  or  of  any  number  into  itself ; 
as  3,  13,  127. 

A  composite  number  is  the  product  of  other  numbers 
greater  than  1 ;  as  4=2X2  ;  12-4X3  ;  150-10X5X3. 


PKIME   NUMBERS. 


117 


Art.  79.— To  resolve  Composite  Numbers  into  Prime 
Numbers  or  Factors. 

Small  composite  numbers  may  be  thus  resolved  by  in- 
spection ;  thus,  the  prime  factors  of  6  are  3  and  2  ;  of  15 
5  and  3. 

The  prime  factors  of  larger  numbers  are  found  by  trial. 

EXAMPLE  1. — What  are  the  prime  factors  in  42  ? 

Process. — Beginning  with  the  least  prime  number,  2,  we 
find  that  it  is  a  factor  in  42,  and  21  another  factor  ;  and  by 
further  trial  we  find  that  3  is  a  factor  in  21,  and  7  another 
factor.  Hence  2,  3,  and  7,  being  all  prime  numbers,  are  the 
prime  factors  in  42. 

BULE. — Divide  the  given  number  by  any  prime  number, 
and  the  quotient  in  the  same  manner,  till  it  is  found  to  be  also 
a  prime  number.  The  several  divisors,  and  the  last  quotient 
will  be  the  prime  factors  of  the  given  number. 

[Any  one  familiar  with  the  multiplication  or  division  table  can 
easily  distinguish  prime  numbers  less  than  144  from  composite  num- 
bers. To  find  other  prime  numbers  less  than  1000,  reference  may  be 
made  to  the  following] 

TABLE  OF  PRIME  NUMBERS. 


2)42 
3)21 

7 


149 

193 

241 

293 

353 

409 

461 

509 

571 

617 

661 

727 

773 

829 

^3 

947 

151 

197 

251 

307 

359 

419 

463 

521 

577 

619 

673 

733 

787 

839 

887 

953 

157 

199 

257 

311 

367 

421 

467 

523 

587 

631 

677 

739 

797 

853 

907 

967 

163 

211 

263 

313 

373 

431 

479 

541 

593 

641 

683 

743 

809 

857 

911 

971 

167 

223 

269 

3171379 

433 

487 

547 

599 

643 

691 

751 

811 

859 

919 

977 

173 

227 

271 

331 

383 

439 

491 

557 

601 

647 

701 

757 

821 

863 

929 

983 

179 

229 

277 

337 

389 

443 

499 

563 

607 

653 

709 

761 

823 

877 

937 

991 

181 

233 

281 

347 

397 

449 

503 

569 

613 

659 

719 

769 

827 

881 

941 

997 

191 

239 

283 

349 

401 

457 

EXAMPLES. 


What  are  the  prime  factors  in 


(2.)  8 
(3.)  10 
(4)  U 


(5.)  21 
(6.)  115 
(7.)  35 


(8.)    32 

(9.)  230 

(10.)  256 


(11.)  288 
(12.)  720 
(13.)  1728 


118  THE   GKEATEST  COMMON  DIVISOR. 

The  Greatest  Common  Divisor. 

Art,  80,— The  Greatest  Common  Divisor  (g.  c.  a.) 

of  two  or  more  numbers  is  the  greatest  number  that  will 
divide  each  of  them  without  a  remainder  ;  as  3  is  the  g. 

c.  d.  of  6  and  9. 

Art,  81, — To  FIND  THE  GREATEST  COMMON  DIVISOR  OF 
ANY  NUMBERS. 

This  may  be  often  done  by  inspection  ;  thus  the  g.  c. 

d.  of  18  and  27  is  evidently  9. 

Or  the  greatest  common  divisor  may  be  found  by  trial. , 

Ex.  1.  What  is  the  g.  c.  d.  of  96,  144,  1728. 

Process.  —We  find  by  trial  that  12  is  a  common     12)96    144    1728 
divisor  of  the  given  numbers  ;   also  that  4  is  a        4^3       J2      144 
common  divisor  of  the  quotients,  and  that  there 
is  no  other  common  divisor.     Since  dividing  by  12 
and  4  is  the  same  as  dividing  by  48,  the  greatest  common  divisor  is  48- 

RULE. — Divide  the  given  numbers  by  any  number  that 
will  divide  each  of  them  without  a  remainder,  and  the  quo- 
tients in  the  same  manner,  till  the  last  common  divisor  is 
found  ;  the  product  of  the  several  common  divisors  will  be  the 
greatest  common  divisor. 

The  products  of  the  prime  factors  common  to  all  the  given  numbers 
will  also  be  the  g.  c.  d. 

The  common  method  of  finding  the  g.  c.  d.,  has  been  to  divide  the 
greater  of  two  numbers  by  the  less,  and  the  last  divisor  by  the  last 
remainder,  till  nothing  remains.  The  last  divisor  is  the  g.  c.  d.  Pro- 
ceed in  the  same  manner  with  the  divisor  thus  found  and  another 
number. 

Ex.  2. — What  is  the  greatest  common  divisor  of  48,  114,. 
132? 

Process.— Any  divisor  of  48  is  a  divisor  of       48^Q^2 
48X2=96.     Therefore  any  divisor  of  48  and 
114  is  a  divisor  of  114  and  96,  also  of  their  18)48(2 

difference,  18.     For  the  same  reason  it  is  a  36 

divisor  of  12,  the  next  remainder,  and  6,  the  ~~12)18(1 

divisor  of  12,  is  the  g.  c.  d.      Proceed  in  the  12 

same  way  with  6  and  132.  "6)12(2 

6)132  12 

22  0 


LEAST   COMMON   MULTIPLE.  119 

This  method  depends  on  the  principle,  tii.it  the  divisor  of  any 
number  is  the  divisor  of  any  product  of  it,  and  (he  divisor  of  any  two 
numbers  is  also  the  divisor  of  their  sum  or  difference. 

EXAMPLES. 

What  is  the  greatest  common  divisor  of 


(3.) 

85 

105 

(7.) 

72 

84 

(11.) 

12 

18 

24 

(40 

90 

152 

(8.) 

84 

96 

(12.) 

18 

32 

40 

(5.) 

100 

150 

(9.) 

98 

112 

(13.) 

44 

66 

88 

(6.) 

108 

114    (10.) 

100 

120 

(14.) 

81 

90 

117 

The  Least  Common  Multiple. 

Art*  82. — The  multiple  of  any  number  is  the  product 
of  that  number  multiplied  by  another  number ;  as  15  i& 
the  multiple  of  5  by  3  or  3  by  5. 

The  Common  Multiple  of  two  or  more  numbers  is  the 
same  or  common  product  of  each  multiplied  by  other 
numbers ;  as,  36  is  the  common  multiple  of  12  by  3  ;  9 
by  4  ;  6  by  6  ;  4  by  9  ;  3  by  12. 

The  Least  Common  Multiple  (1.  c.  in. )  of  two  or  more 
numbers,  is  the  least  common  product  of  each  by  other 
numbers  ;  as,  24  is  the  1.  c.  m.  of  12  (by  2)  ;  8  (by  3; ; 
6  (by  4). 

Art.  83. — To  FIND  THE  LEAST  COMMON  MULTIPLE  OF  Two 
OR  MORE  NUMBERS. 

Ex.  1.  What  is  the  1.  c.  m.  of  4,  6,  12. 

Process. — By  inspection  we  find  that  12  is  a  common  multiple  of 
4  (by  3  ,  and  6  (by  2),  and  there  can  be  no  less  multiple  of  itself  'r 
therefore  it  is  the  least  common  multiple  of  the  given  numbers. 

RULE  I. —  When  the  largest  given  number  is  a  common 
multiple  of  the  others,  it  is  the  least  common  multiple. 

Ex.  2.  What  is  the  least  common  multiple  of  6,  8,  12  ? 
Process.-  -By  inspection  we  find  that  12X2  will  be  a  common 
multiple  of  6  (X4)  and  8  (X3).     Therefore  it  is  the  1.  c.  m. 

RULE  H. —  When  the  largest  number  is  not  already  a  com- 
mon multiple  of  the  othei*s,  multiply  it  by  the  least. number 


120  LEAST   COMMON   MULTIPLE. 

thai  ivill  evidently  make  it  such,  and  the  product  will  be  the 
least  common  multiple. 

If  it  is  not  known  by  what  number  the  largest  given  number  must 
be  multiplied  to  make  it  a  common  multiple  of  the  others,  the  1.  c.  m. 
may  be  found  by  the  following  method. 
The  common  multiple  of  6,  8,  12  (Ex.  2)  is 

6X8X12=  ) 

8X6X12=  f  576.  But  6,  a  common  factor  in  two  of  the  numbers 
12X6X  M 

(6  and  12)  ,  and  4,  a  common  factor  in  8  and  12,  are  common  factors 
in  the  multipliers  of  the  given  numbers,  and  therefore  may  be  can- 

6X2X2-  ) 

celled,  leaving    8X1X3=  Y  24  the  1.  c.  m. 
12X1X2=  ) 

Since  each  product  is  the  same,  it  is  not  necessary  to  repeat  the 
process,  and  it  makes  no  difference  whether  the  common  factors  be 
cancelled  in  the  multipliers  of  the  given  numbers,  or  in  the  numbers 
themselves  ;  hence 

2o?  Process.  —  Cancel  the  6  because  it  is  a  factor  ( 
in  12,  and  4  the  greatest  factor  in  8,  because  it  is  J  0       $     12 
also  a  factor  in  12.    Then  multiply  the  given  num-  (  2X12=24  Ans. 
ber,  12,  by  the  factor  2,  and  the  product,  24,  is  the 
least  common  multiple. 

RULE  IH.  —  Cancel  any  number  or  greatest  factor  that  is 
also  a  factor  in  another  number,  and  the  product  of  the  re- 
maining numbers  and  factors  will  be  the  least  common 
multiple. 

Sometimes  two  factors  in  the  same  number  may  be  omitted  when 
they  are  also  factors  in  different  numbers. 

If  there  is  no  common  factor,  the  product  of  the  given  numbers  will 
be  the  least  common  multiple. 

The  common  method  has  been  the  following  : 

2)6  8  12 
2)3  4  6 
3)3  2  3 

TT1 

Ans. 


RULE  IV.  —  Divide  the  numbers  by  any  prime  number  that 
will  divide  two  or  more  of  them  without  a  remainder  ;  place  the 
quotients  and  undivided  numbers  in  another  line  ;  divide  these 
also  in  tfie  same  manner,  and  continue  the  process  till  no  two 


THE   GREATEST   COMMON   DIVISOR.  121 

numbers  can  be  thus  divided.     Then  multiply  together  all  the 
divisors  and  undivided  numbers. 


EXAMPLES. 

What  is  the  1.  c.  m.  of 


(3.)    4,  8,  9? 

(4.)     6,  9,  12  ? 

(5.)     9,  15,  18  ? 

(6.)  10,  20,  30? 


(7.)  5,  10,  12,  24? 

(8.)  6,  12,  14,  28? 

(9.)  8,  14,  21,  30? 

(10.)  9,  11,  22,  27? 


Art.  84, — Promiscuous  Examples  in  Properties  of  Numbers. 

1.  What  are  the  prime  factors  in  24  ?  35  ?  196  ? 

2.  What  is  the  g.  c.  d.  of  9,  18,  24,  30  ? 

3.  What  is  the  1.  c.  m.  of  8,  12,  20,  32  ? 

4.  What  are  the  prime  factors  in  14  ?  26  ?  34  ? 

5.  What  is  the  g.  c.  d.  of  21,  28,  35,  70  ? 

6.  What  is  the  1.  c.  m.  of  9,  11,  18,  22  ? 

7.  There  are  three  rooms  respectively,  12,  18  and  24  feet 
wide  ;  how  wide  may  be  the  widest  oil-cloth  that  will  exactly 
fit  them  all  without  being  cut  ? 

8.  There  are  three  horses  running  a  circuit ;  the  first  can 
complete  it  in  10  minutes,  the  second  in  12,  and  the  third  in 
15  ;  if  they  start  together  and  keep  running,  how  long  will  it 
be  before  they  will  be  together  again  ? 

9.  There  is  a  garden  the  sides  of  which  are  respectively,  112, 
126,  140  and  154  feet  ^what  must  be  the  length  of  the  longest 
boards  that  will  fence  it  without  being  cut  ? 

10.  Three  ferry  boats,  starting  together,  are  making  regular 
trips  to  different  points  and  back  ;  the  first  in  20,  the  second 
in  25,  and  the  third  in  30  minutes  ;  how  long  -will  it  be  before 
they  will  all  return  together  ? 


FRACTIONS. 

Art.  85. — A  Fraction  is  one  or  more  equal  parts  of  a 
number  or  thing,  and  expresses  division. 


122  FRACTIONS 

If  a  number  or  thing  is  divided  into  two  equal  parts, 
one  of  them  is  called  one-half ;  if  divided  into  three 
equal  parts  one  of  them  is  one-third,  two  of  them  two- 
thirds  ;  if  divided  into  four  equal  parts,  one  of  them  is 
called  one-fourth,  or  one-quarter. 

If  the  number  thus  divided  is  not  expressed,  it  is  un- 
derstood to  be  one,  or  a  single  thing  ;  thus  in  the  line 
AE,  A  o  is  one-half,  A  B  one-quarter  ;  A  D  three-quarters  ; 
A  F  one-third  ;  A  a  two-thirds  of  the  line  AE. 


E 


Art.  86. — Fractions  are  of  two  kinds,  Common  and 
Decimal. 

Common  fractions  are  such  as  express  any  number  of 
equal  parts ;  as,  one-hall,  two-thirds. 

Decimal  fractions  are  such  as  express  only  one  or  more 
of  ten  equal  parts,  or  ten  times  ten,  &o.;  as,  one-tenth,, 
three-hundredths,  five-thousandths,  &c. 


Common  Fractions. 

Art.  87. — Common  fractions  are  usually  written  with 
two  numbers,  one  the  above  other,  with  a  line  between 
them ;  as,  J,  one-half ;  f ,  two-thirds. 

The  number  above  the  line  is  called  the  Numerator, 
and  the  number  below  the  line  the  Denominator  $  both 
together  are  called  the  Terms  of  a  fraction. 

The  Denominator  shows  the  number  of  equal  parts  in- 
to which  anything  is  divided,  and  corresponds  with  the 
divisor  in  division  ;  also  gives  name  to  the  fraction. 


FRACTIONS.  123 

The  Numerator  shows  how  many  of  the  equal  parts  are 
taken,  or  it  may  be  a  number  divided  by  the  denomina- 
tor, and  the  same  as  the  dividend  in  division. 


.  —  In  the  fraction  f,  the  denominator  (3)  shows  that 
something  is  considered  as  divided  into  three  equal  parts,  and  the 
numerator  (2)  shows  that  there  are  two  such  parts,  or  two  things 
thus  divided.  The  traction  expresses  either  two-thirds  of  one,  or 
one-third  of  two  ;  as,  one-third  of  two  dollars,  or  two-thirds  of  one 
dollar. 

Artt  88,  —  Common  fractions  are  usually  divided  into 
Simple,  (either  proper  or  improper,)  Compound,  Complex, 
and  Mixed  Numbers. 

A  Simple  fraction  has  but  one  numerator  and  one  de- 
nominator, both  whole  numbers.  It  is  called  a  Proper 
fraction  when  its  numerator  is  less  than  its  denominator, 
and  Improper  when  its  numerator  is  equal  to  or  greater 
than  its  denominator  ;  as,  1,  (proper)  f  ,  or  f-  ,  (improper.  ) 

A  Compound  fraction  is  a  fraction  of  a  fraction  ;  as 
fof  f. 

A  Complex  fraction  is  one  whose  numerator  or  denomi- 

f      3 
nator  is  a  fraction  or  mixed  number  ;  as,  |y,    ^i- 

A  Mixed  Number  is  a  whole  number  and  fraction 
written  together  ;  as  3f  . 

It  will  be  found  convenient,  also,  to  divide  common 
fractions  into  two  other  kinds,  Like  and  Unlike. 

Like  fractions  are  such  as  have  the  same  name  or  com- 
mon denominator  ;  Unlike,  such  as  have  different  denom- 
inators ;  as,  i,  f  ,  I  are  Like  fractions  ;  f,  £,  f-  are  Unlike. 

Art.  89.  —  The  Value  of  a  fraction  is  the  quotient  of  the 
numerator  divided  by  the  denominator  ;  as,  the  value  of 
f  is  2  ;  of  f  is  1.  The  value  of  a  proper  fraction  is  less 
than  1,  and  therefore  can  only  be  expressed  in  the  form 
of  a  fraction  ;  as  f  . 


124  FRACTIONS. 

The  value  of  the  equal  parts  of  a  fraction  depends  on 
the  denominator,  or  the  number  of  parts.  The  less  the 
denominator  the  greater  the  value  of  each  part  ;  the 
greater  the  denominator  the  less  the  value  of  each  part  ; 
thus  i  is  more  than  i.  f  than  I. 

Art*  90«  —  Since  the  numerator  of  a  fraction  corres- 
ponds with  the  dividend  in  division,  the  denominator  with 
the  divisor,  and  the  value  with  the  quotient,  the  follow- 
ing propositions  are  of  frequent  application  in  fractions. 

I.  Multiplying  the  numerator  or  dividing  the  denominator 
by  any  number,  multiplies  the  value  of  the  fraction  by  that 
number  ;  thus  — 

;6X4==8  (2X4).  == 


II.  Dividing  the  numerator  or  multiplying  the  denominator 
by  any  number,  divides  the  value  of  the  fraction  by  that  num- 
ber ;  thus  — 

2==2 


HE.  Multiplying  or  dividing  both  numerator  and  deno- 
minator by  the  same  number  does  not  alter  the  value  of  a 
fraction;  thus  — 


EXEBCISES. 

Read,  name  the  kind,  and  explain  the  terms,  of  the  follow- 
ing fractions  : 

K  t,  f  ,  f  ,  i-  of  f  ,  3i,  f  ,  f  ,  |,    I  ,  JgS  I,   6f  ,  If,  f  of  ?. 

4  Of 

Also,  mention  some  of  the  same  that  are  like,  and  others 
that  are  unlike. 

Write  and  describe  the  following  fractions  : 

Three-fifths,  seven-ninths,  nine-fourths,  eleven-quarters, 
two-thirds  of  four-ninths,  nine  and  a  half,  five-sevenths,  sev- 


FRACTIONS.  125 

en-thirds,  four-fifths  of  seven-eighths,  six  and  three-quarters, 
3-sevenths,  5-ninths,  8-etevenths,  10-  twelfths. 

Bead  the  following  in  order  according  to  the  value  of  their 
equal  parts,  beginning  with  the  greatest.  Kepeat  the  same, 
beginning  with  the  least  : 

t,  A,  i,  i  A.  i,  -/o,  I,  H,  I,  A,  f,  i  A, 

Write  a  fraction  of  each  kind  repeatedly  till  it  can  be  done 
correctly. 

MENTAL  EXERCISES. 

If  an  apple  or  anything  is  divided  into  two  equal  parts, 
what  is  each  part  called  ?  If  divided  into  three  equal  parts  ? 
4?  5?  6?  7?  8?  9?  10?  11?  12?  13?  15?  18? 
20?  25?  50?  65?  84?  100? 

If  an  orange  or  anything  is  divided  into  12  equal  parts,  what 
is  one  of  them  called  ?  3  of  them  ?  6?  9?  4?  7?  10? 
12? 

Which  is  the  greater,  -^  or  £  ?     Why  ?    f  or  f  ?    |  or 
or?       or?         or?  or 


Art.  91.  —  REDUCTION  or  FRACTIONS. 

Reduction  of  fractions  is  changing  their  form  without 
altering  their  value  ;  as,  |=|,  t=|,  f=2|. 

Fractions  are  reduced  to  their  simplest  form  when  they 
are  reduced  to  simple  and  proper  fractions  in  their  lowest 
terms,  whole  or  mixed  numbers  ;  thus  the  simplest  form 
of  |f  is  i  ;  of  f  is  2  ;  of  |  is  1|  ;  f  of  Hs  &. 

CASE  I. 

Artt  92.  —  To  reduce  a  fraction  to  its  lowest  terms  or 
simplest  form. 

A  fraction  is  in  its  lowest  terms,  when  its  value  is  expressed  by  the 
least  numbers  possible  ;  thus  -&  reduced  to  its  lowest  terms  is  £." 

EXAMPLE  1.  —  Reduce  if  to  its  lowest  terms. 

The  fraction  |-|  can  be  reduced  to  lower  terms  because  both  its 
numerator  and  denominator  can  be  divided  by  2  and  3,  or  6,  which 
(Prop.  HI.  Art.  90)  does  not  alter  its  value. 


126 


FARCTIONS. 


Process.  —  Dividing  both  terms  by  3,  the  result  is  3)18_2)6_3 
f,  which  may  also  be  divided  by  2,  the  result  being  -54—  0—7 

I,  the  lowest  terms.     Or  dividing  £f  by  6  the  result 
is  the  same.  or  6)_18=3 

24    4 

RULE.  —  Divide  the  numerator  and  denominator  by  any 
number  that  will  divide  them  both  without  a  remainder,  and 
continue  dividing  till  the  lowest  terms  are  found. 

Or  divide  both  terms  by  their  greatest  common  divisor. 


EXAMPLES. 


(2.) 
(3.) 

(5.) 


M 

if 


(6.) 
(7.) 

(8.) 
(90 


M 

M 

if 
« 


(ii.) 

(12.) 


Hi 

m 


(15.) 

(160 

(17.) 


tff 


CASE  H. 

Art.  93.  —  To  reduce  an  improper  fraction  to  a  whole  or 
mixed  number,  or  its  simplest  form. 

MENTAL  EXEBCISES. 

In  2  half  dollars  or  2  halves  (•§  )  of  one  dollar,  how  many 
dollars?  Inf?  |?  f?  f?  f?  tf?  f?  |?  f?  f? 
f?  §?  V?  |?  |?  |?  |?  V?  f?  f?  f?  ^? 


EXAMPLES  FOR  THE   SLATE. 

EXAMPLE  18.  —  Reduce  ^-  to  a  whole  number. 

Process.—  Since  the  denominator  (3)  shows  the  number  of  27 
equal  parts  in  1,  and  the  numerator  (27)  how  many  such  3 
parts  there  are,  the  fraction  ^  is  as  many  times  1  as  3  is 
contained  times  in  27,  which  are  9  times.  Therefore 
a  whole  number. 


2.  Reduce 
Process.—  18 


to  a  mixed  number. 


_ 
~~3£  a  mixed  number. 

RULE.—  Divide  the  numerator  by  the  denominator,  and  the 
quotient  will  be  the  whole  or  mixed  number. 


FRACTIONS. 


127 


(19.)  Jj 
(20.)  \ 

(21.)  \ 

34.  In 

35.  In 

36.  In 

37.  In 


EXAMPLES. 

(22.)  J>£  I  (25.)  ag.  (28.) 
(23.)  *£  (26.)  -^  (29.) 
(24.)  ^  I  (27.)  3£  (30.) 

of  a  mile,  how  many  miles  ? 

of  a  pound  how  many  pounds  ? 

of  a  yard  how  many  yards  ? 

of  a  bushel  how  many  bushels  ? 

CASE  HI. 


(31.) 
(32.) 
(33.) 


Art*  94.  —  To  reduce  whole  or  mixed  numbers  to  improper 
fractions. 

MENTAL   EXERCISES. 

In  1  apple  or  anything  how  many  halves  ?  [Let  the  answer 
be  first  oral,  then  written.]  How  many  thirds  ?  fourths  ? 
quarters  ?  fifths  ?  &c. 

In  3  apples  ?  5  ?  7  ?  8  ?  &c. 

In  \yz  apples  how  many  halves?  2^?  4^?  6^?  7}£? 

In  IK  apples  how  many  thirds  ?  2%?  3^?  5%?  7^? 

UK?  12%? 

In  1%  apples  how  many  quarters  ?  2^?  3%?  5^? 


5 

33 
-=- 


EXAMPLES  FOE   THE   SLATE. 

38.  Beduce  64  to  an  improper  fraction. 

Process.  -Since  1=£  6=6  times  f=AA  to  which  add  f 
and  the  sum  will  be  ^  u4ns.  Or, 

Since  there  are  f  in  1,  in  6  there  are  5  times  as  many  fifths 
as  there  are  times  1,  and  5  times  6  are  30.  Therefore  6=^- 
to  which  add  f,  and  the  sum  will  be  a5a,  Ans. 

EXAMPLE  39.  —  Beduce  8  to  an  improper  fraction. 
Process.  —Since  1=1,  8=8  times  |=f. 

RULE.  —  To  reduce  a  mixed  number  to  an  improper  fraction, 
multiply  the  whole  number  by  the  denominator  of  the  frac- 
tion, add  the  numerator,  and  write  the  denominator  under 
the  sum. 

To  reduce  a  whole  number  to  an  improper  fraction,  write 
1  under  it  with  a  line  between  them. 


128  FRACTIONS. 


EXAMPLES. 


(40.)  6J 
(41.)  7| 
(42.)  8} 


(43.)      9f 
(44.)     10 
(45.)     Ill 


(46.)  12f 
(47.)  15f 
(48.)  18| 


(49.)     25 
(50.)    30^ 
(51.) 


52.  In  3£  pounds,  how  many  fifths  of  a  pound  ? 

53.  In  11|  hours,  how  many  thirds  of  an  hour  ? 

54.  In  12£  rods,  how  many  ninths  of  a  rod  ? 

CASE  IV. 

Art.  95.  —  To  reduce  unlike  fractions  (having  different  de- 
nominators,) to  like  fractions  (having  a  common  denom- 
inator. ) 

EXAMPLE  55.  James  has  %  of  an  orange,  John  %,  Henry  %; 
how  can  these  unequal  parts  be  divided  into  equal  parts  ? 

Since  the  denominator  of  a  fraction  shows  the  number  of  equal 
parts  into  which  anything  is  divided,  fractions  having  different  de- 
nominators must  be  reduced  to  a  common  denominator,  that  all  the 
parts  may  be  equal. 

Process.  —To  reduce  £  f  f  to  a  common  denominator,  we  multiply 
the  numerator  and  denominator  of  each  fraction  by  all  the  other  de- 
nominators which  (Prop.  III.  Art.  90)  does  not  alter  their  value  ;  thus 


The  fractions  £,  f,  f-,  are  thus  reduced  to  equivalent  fractions  £f  , 
£f  ,  £f  ,  having  a  common  denominator.  Therefore  if  £,  f  ,  f  ,  of  an 
orange  be  divided  respectively  into  12,  16,  18  equal  parts  all  the 
parts  will  be  equal,  and  each  part  will  be  -^  of  an  orange. 

RULE  I.  —  Multiply  the  numerator  and  denominator  of 
each  fraction  by  all  the  other  denominators. 

Since  the  denominators  thus  found  will  be  the  same,  after  one  is 
found  it  may  be  taken  for  the  common  denominator. 

EXAMPLES. 

(56.)     iff  (59.)     fff  (62.)     J    f    f 

(57.)     if?  (60.)     $f»  (63.)     f    J    A 

(58.)     Ill  (61.)     I    |   A  (64.)     |    f    f 

Art.  96.  —  To  reduce  unlike  fractions  to  equivalent  like  frac- 
tions having  the  least  common  denominator. 


FRACTIONS.  129 

The  fractions  £,  f,  £  (Ex.  55, )  reduced  to  a  common  denominator, 
are  £f ,  £f ,  if ,  but  these  may  be  reduced  to  lower  terms,  so  that  the 
common  denominator  will  be  less,  as  •&,  f2 ,  & .  12  is  the  least  common 
denominator,  because  the  fractions  cannot  be  reduced  to  lower  terms 
and  still  have  a  common  denominator. 

To  find  the  least  common  denominator  of  £,  f ,  f ,  we  multiply  the 
numerator  and  denominator  of  the  fraction  £  by  6,  f  by  4,  £  by  3, 
because  we  observe  that  the  fractions  thus  multiplied  will  have  a 
common  denominator,  the  least  that  can  be  found,  and  become  ,d2, 
A,  A,  Ans. 

RULE  n. — Multiply  the  numerator  and  denominator  of  each 
fraction  by  the  least  number  that  mil  produce  a  common  de- 
nominator. 

If  such  a  number  cannot  be  conveniently  found  by  in- 
spection, find  the  least  common  multiple  of  the  denomin- 
ators, and  divide  it  by  the  denominator  of  each  fraction. 
The  least  common  multiple  will  be  the  least  common  de- 
nominator, therefore  the  denominators  need  not  be 
multiplied. 

Before  reducing  fractions  to  a  common  denominator  reduce  them 
to  their  lowest  terms. 

EXAMPLES. 

(65.)     £    f    |  (68.)     |     i    H  (71.)     f     f     i 

(66.)     iff  (69.)     i    I    A  (72.)     §    f    A 

(67.)     iff  (70.)     f    f     |  (73.)     iff 

When  there  are  no  two  denominators  of  the  fractions  to  be  reduced, 
which  can  be  divided  by  the  same  number,  the  first  rule  for  finding 
the  common  denominator  must  be  used. 

EXAMPLES  UNDER  BOTH  BULES. 

(74.)  Vi    *  (77.)     I    |    M  (80.)     |     |    £ 

(75.)     f    |    f  (78.)     I    |    i|  (81.)    A    &   & 

(76.)     1    f    |  (79.)     i   ^   A  (82.)     |    ^    | 

The  object  of  reducing  fractions  to  a  common  denominator,  is  to 
prepare  them  for  addition  and  subtraction. 

Art,  97.— Reduction  of  Compound  Fractions  is  prop- 
erly included  in  Multiplication  of  Fractions,  and  Reduc- 
tion of  Complex  Fractions  in  Division  of  Fractions. 


130  FRACTIONS. 


Art,  98.— Promiscuous  Examples  in  Reduction  of 
Fractions. 

1.  Beduce  ££  to  its  lowest  terms,  or  simplest  form. 

2.  Reduce  ^  to  a  mixed  number,  or  its  simplest  form. 

3.  Eeduce  9§  to  an  improper  fraction. 

4.  Reduce  ^,  it,  ^,  §  to  like  fractions  or  a  common  denomin- 
ator. 

5.  Reduce  ^§^  to  its  simplest  form. 

6.  Reduce  12|  to  an  improper  fraction. 

7.  Reduce  ^  to  its  simplest  form. 

8.  Reduce  f ,  f ,  -|,  4  to  like  fractions. 

9.  Reduce  \8  to  a  mixed  number. 

10.  Reduce  ff  to  its  lowest  terms. 

11.  Reduce  13f  to  an  improper  fraction. 

12.  Reduce  f ,  -&,  f ,  3  to  like  fractions. 

13.  Reduce  f  f  to  its  simplest  form. 

14.  Reduce  ^  to  its  simplest  form. 

15.  Reduce  14^  to  an  improper  fraction. 

16.  Reduce  f ,  T%,  f ,  6  to  like  fractions. 

17.  Reduce  ff  to  its  lowest  terms. 

18.  Reduce  f  f  to  a  whole  number. 

19.  Reduce  15^T  to  an  improper  fraction. 

20.  Reduce  f ,  f ,  f ,  5  to  like  fractions. 

21.  Reduce  f ,  $,  ^,  f  to  like  fractions. 

22.  Reduce  15f ,  to  an  improper  fraction. 

23.  Reduce  ^  to  its  simplest  form. 

24.  Reduce  ^|  to  its  simplest  form. 

25.  Reduce  f ,  f ,  f ,  2  to  like  fractions. 

26.  Reduce  f ,  f ,  f ,  1  to  like  fractions. 

27.  Reduce  3/  to  a  mixed  number. 

28.  Reduce  9f  to  an  improper  fraction. 

29.  Reduce**^  to  a  whole  number. 

30.  Reduce  £,  f ,  -$y,  7  to  like  fractions. 

31.  Reduce  f  £  to  its  lowest  terms. 

32.  Reduce  £,  ^,  f ,  3  to  like  fractions. 


ADDITION   OF   FRACTIONS.  131 


33.  Eeduce  16^  to  an  improper  fraction. 

34.  Keduce  ^fi  to  a  mixed  number. 

35.  Reduce  f ,  f ,  f ,  -^  to  like  fractions. 

36.  Reduce  f ,  f ,  ^  4  to  like  fractions. 

37.  Reduce  f|f|  to  its  lowest  terms. 

38.  Reduce  *£f£  to  its  simplest  form. 

39.  Reduce  f ,  f ,  fj,  3  to  like  fractions. 

40.  Reduce  f,  1 1,  -^  to  like  fractions. 

41.  Reduce  |,  •&,  -§ ,  §  to  like  fractions. 


Addition  of  Fractions. 

Art.  99.— Addition  of  Fractions  is  finding  their  sum. 
EXAMPLE  1.— Add  £,  f,  f ,  f . 

Since  £,  f,  <fcc.,  are  like  fractions  they  can  be  1  fifth, 

added  by  adding  their  numerators,  the  same  as  1  cent,  2     « < 

2  cents,  &c.  3     " 

Process.—  i+f=£  and+f=f  and+^^2  Arcs.  _4_    " 

10     »     J#=2 
Ex.  2.  Addi  |,|. 

Since  |,  |  and  |  are  unlike  fractions,  they  cannot  be  added  as  they 
are,  and  make  f,  f ,  or  f  any  more  than  1  dollar  2  cents  and  3  mills  can 
be  added  in  one  column  and  make  6  dollars  6  cents  or  6  mills. 

Process.— Eeduce  £,  |,  f  to  like  fractions,  £f,  £f,       12        6 
if,  or  *,  &,  4,  and  add  as    above  i|+M+if=      16        8 

M==!i=ittOT«f8J>    H=IH. 

^S=i« 

^ns.    24~~12 

RULE. — Reduce  the  fractions  to  like  fractions  or  a  common 
^Denominator,  add  their  numerators,  and  under  the  sum  write 
the  common  denominator. 

1.  Whole  and  mixed  numbers  may  be  reduced  to  improper  fractions 
and  then  added  like  other  fractions  :  but  it  better  to  add  the  fractional 
parts  separately,  and  the  whole  numbers  to  the  result. 

2.  After  adding,  reduce  the  sum  to  its  simplest  form. 


132  SUBTRACTION  OF  FRACTIONS. 


EXAMPLES. 


(3.)  Add}       J 


(10.) 

(5.; 


(9.)  Add    f 


(12.)     «    3^      f      2} 
(13.)     "    2|      31        f      1 
(14.)     "     &        i        A      3 


(7.)     "     f      2|        f 
(80     "     f        f        J 

15.  How  many  dollars  will  pay  for  a  coat  worth  $12},  pants 
$9f ,  vest  $41,  hat  $5},  and  a  pair  of  boots  $6£  ? 

16.  How  many  yards  in  five  pieces  of  cloth  measuring  27^, 
25f ,  261,  -241,  22$  ? 

17.  How  many  pounds  of  tea  in  six  packages  weighing  6§, 
54,  3,  4f ,  21,  i£  ?  pound  ? 

18.  How  many  pounds  of  butter  in  four  tubs  weighing  18|, 
16},  20ft,  19f  ? 

19.  How  many  hundred  weight  of    sugar  in  five  barrels 
weighing  2},  l^fc,  2f,  1}§,  2f  cwt? 

20.  How  many  tons  in  four  loads  of  coal  weighing  1^,  1|, 
*M,  l^W  tons  ? 


Subtraction  of  Fractions. 

Art.  100,— Subtraction  of  Fractions,  is  finding  the 
difference  between  two  fractions  or  a  fraction  and  a  whole 
number. 

Ex.  1.  From  f  take  f . 

Since  f  and  f  are  like  fractions,  they  can  be  subtracted  by  taking 
the  less  numerator  from  the  greater,  the  same  as  taking  3  cents  from 
4  cents. 

Process. — f — f=}  Ans. 

Ex.  2.  From  f  take  }. 

Since  f  and  $  are  unlike  fractions,  they  cannot  be  subtracted  as  they 
are,  any  more  than  1  cent  can  be  taken  from  3  dollars  and  leave  2  dol- 
lars or  2  cents. 

Process.  •-  Eeduce  f,  £  to  like  fractions  ja2,  ^,  and  substract  ^  from' 
fa.  The  remainder,  &,  is  the  answer. 


SUBTRACTION   OF  FRACTIONS.  133 

RULE. — Reduce  the  fractions  to  like  fractions  or  a  common 
denominator,  subtract  the  numerators,  and  under  the  remain- 
der write  the  common  denominator. 

1.  After  subtraction  reduce  the  remainder  to  its  simplest  form. 

2 .  Whole  or  mixed  numbers  may  be  reduced  to  improper  fractions 
and  then  subtracted,  but  it  is  better  to  subtract  the  fractional  parts 
separately,  thus 

Ex.  3.  From  4  subtract  f . 

Process. — Since  there  is  no  fraction  from  which  to  take  f,  4 

take  it  from  1  (a  part  of  the  4)=£.    £ — f=f  and  since  1  has  £ 

been  used  in  the  operation,  carry  as  in  subtraction  of  whole  g? 
numbers. 

Ex.  4.  From  8^  subtract  3f . 

Process. — Reduce  ^  and  f  to  like  fractions,  -fa,  ££,  and  since  8 

££  cannot  be  subtracted  from-/4-,  addl  (a  part  of  the  8)=f£to  3 

5\,  and  from  their  sum  |  f  take  ££,  and  the  remainder  will  be  ^\,  "£TT 
carry  1,  &c. 


EXAMPLES. 


(5.)  From     f  subtract  f 

(6.)       "        |        "  | 

(7.)       "      *       "  f 

(8.)       »       M        "  f 

(9.)       "       M        "  A 

(10.)       "      fts      "  f 


(11.)  From  4|  subtract  2£ 

(12.)  "   5f   "    31 

(13.)  "  lOf    "    3T 

(14.)  "   9^   "   4f 

(15.)  "14    "    Cf 

(16.)  "  18f    "    9^ 


17.  If  a  man  have  $6,  and  spend  $3f ,  how  many  dollars  will 
lie  have  left  ? 

18.  If  a  piece  of  cloth  contain  27£  yds.,  and  Of  yds.  be  cut 
off,  how  many  yards  will  be  left  ? 

19.  If  I  buy  25  Ibs.  of  sugar  and  use  12-j6^  Ibs.,  how  much 
will  be  left  ? 

20.  If  a  man  on  a  journey  of  65^  miles,  has  traveled  48f 
miles,  how  many  more  miles  has  he  to  travel  ? 

21.  If  there  are  37£  gals,  of  wine  in  a  cask,  how  many  will 
be  left  after  15§  gals,  shall  have  been  drawn  from  it  ? 

•     22.  If  a  boy  study  5|  hours  and  play  2§  hours,  how  many 
more  will  he  study  than  play  ? 


134  MULTIPLICATION   OF  FRACTIONS. 

Multiplication  of  Fractions. 

CASE  I. 
Art.  101  1  —  Multiplying  a  fraction  by  a  fraction. 

Multiplying  any  number  by  £  is  finding  £  of  it,  which  is  the  same 
as  dividing  it  by  2  ;  multiplying  by  %  is  finding  ^  of  it,  the  same  as 
dividing  it  by  3,  and  £  of  a  number  multiplied  by  2,  gives  f  of  it,  &c. 

EXAMPLE  1.—  Multiply  f  by  f  . 

Process.  -5_X-  =-•    Therefore  *LX-  =  ^=-    , 

8       4     32  8       4      32         32    !LyJ?  =  15 

(Prop.  I.  II.  Art  90.)  84      32 

Ex.  2.  Multiply  f  by  ^. 

^01 

Process.  —  Cancel  4  and  9,  factors  common  to  the  nu-  —  X  —  =~7 
merators  and  denominators,  before  multiplying.  9  ^9  4 

4 

CASE  n. 

Art.  102.  —  Multiplying  a  fraction  by  a  whole  number,  or  a 
whole  number  by  a  fraction. 
Ex.  3.  Multiply  %  by  3. 

Process.—  3  times  £=V  or  f  (P.  I.  Art.  90)=2£. 
Ex.  4.  Multiply  3  by  £. 

Process.—  3X^=1  or  |  ;  therefore  3X1=  V  or  £=2£. 
The  product  of  |X^  is  the  same  as  that  of  3X$,  and  since  3=f 
this  case  may  be  included  in  Case  I. 

CASE  HI. 

Art.  103.  —  Multiplying  mixed  numbers. 
Ex.  5.  Multiply  4§  by  3|. 
1st  Process.—  4|X3* 
2d  Process.  —= 


4X3=      12 
The  products  added  are  17|^.  Ans. 

By  the  first  process  the  mixed  numbers  are  reduced 


REDUCTION  OF  COMPOUND  FRACTIONS. 


135 


to  improper  fractions  and  then  multiplied  as  in  Case  I.  ; 
by  the  second  process,  the  whole  numbers  and  fractions 
are  multiplied  as  in  Case  EL 

KULE.  —  Reduce  whole  or  mixed  numbers  to  improper  frac- 
tions, cancel  all  factors  common  to  the  numerators  and  the  de- 
nominators, then  multiply  the  remaining  numerators  together 
for  a  new  numerator  and  the  remaining  denominators  for  a 
new  denominator. 

A  whole  number  may  be  multiplied  as  the  numerator  of  an  im- 
proper fraction  without  being  reduced. 

Art,  104,—  Reduction  of  Compound  Fractions  is  pro- 

perly included  in  multiplication  of  fractions  : 

f  of  f=fxfHr£  ;  f  of  f  of  2jc=fxiX¥=H=i&- 

MENTAL   EXERCISES. 

If  an  apple  cost  ^  cent,  what  will  ^  of  an  apple  cost  ?    §  ?   £  ? 


Process.—  If  an  apple  cost 
which  is  |  cent. 


cent,  i  of  an  apple  will  i  of  ^  cent, 


At  £  cent  each,  what  will  2  apples  cost  ?4?5?8?11?12? 
If  an  orange  cost  4  cents,  what  will  ^  of  an  orange  cost  ?  f  ? 

f  ?   f?   £? 

What  will  3  plums  cost  at  J  cent  each?  4?  5?  6?  8?  11?  12  ? 
If  a  melon  cost  6  cents,  what  will  ^  a  melon  cost  ?    f  ?    §  ? 

i?   f  ?    A? 
If  a  peach  cost  f  of  a  cent,  what  will  £  of  it  cost  ?    £  ?    §  ? 

f?    f?   I? 

EXAMPLES  FOB  THE   SLATE,    ETC. 

Multiply — 

(13.)     6    by    | 


(6.)  4    by    f 

(7.)  |  by  3 

(8.)  fby    | 

(9.)  31  by  4 

(10.)  5    by2f 

(11.)  6|  by  3± 
(12.)iof3by2J 


{14.) 
(15.) 
(16.) 
(17.) 
(18.) 
(19.) 


± 

4|by5 
6    by  7§ 
8^by5| 
7£byfof6 


(20.) 
(21.) 


8    by    f 
i  by  16 

(22.)      A  bJ   A 
(23.)     6|  by  3 


(24.) 
(25.) 


7    by4f 
9f  by  6f 


136  DIVISION   OF  FRACTIONS. 

27.  At  $|  a  bushel,  what  will  6  bushels  of  potatoes  cost  ? 
13?   27? 

28.  At  $5  a  bushel,  what  will  -|  bushel  of  clover  seed  cost  ? 

f?  f? 

29.  At  $f  a  bushel,  what  |  of  a  bushel  of  corn  cost  ?   f  ?  £  ? 

30.  At  $4£  a  bushel,  what  will  2  bushels  of  timothy  seed 
cost?    3?    5? 

31.  At  $4  a  bushel,  what  will  3f  bushels  of  timothy  seed 
cost?    5§?    6f? 

32.  At  $4£  a  bushel,  what  will  2|  bushels  of  timothy  seed 
cost?   4£?   6f  ? 

33.  At  $6£  a  bushel,  what  will  f  of  3|  bushels  of  clover  seed 
cost? 


Division  of  Fractions, 

CASE  I. 

Art.  105. — Dividing  a  fraction  by  a  whole  number. 
Ex.  1.  Divide  f  by  3. 
Process.—  H-3=*  or  ft  (P.  II.  Art.  90.) 

RULE. — Divide  the  numerator  by  the  whole  number  when  it 
can  be  done  without  a  remainder,  otherwise  multiply  the  de- 
nominator by  the  whole  number. 

A  mixed  number  may  be  reduced  to  an  improper  fraction  and  then 
divided  ;  or  the  integral  part  may  be  divided  first,  and  the  remainder 
afterwards  ;  thus  9^-4-4=^=^fi  or  2  and  a  remainder  1£  which  di- 
vided by  4  equals  f-^-4=f . 


(2.)  Divide  £  by  2 
(3.)  Divide  f  by  3 
(4)  Divide  ^  by  4 


EXAMPLES. 

(5.)  A  by  5 
(6.)  if  by  6 
(7.)  V  by  100 


(8.)  if  by  7 
(9.)  ^  by  8 
(10.)  f£  by  8 


FRACTIONS. 


137 


CASE  II. 

Art.  106. — Dividing  by  a  fraction. 
Ex.  11.— Divide  f  by  f . 

Process. — -|-^1=£,  and  £  is  contained  in     5      2      5      $      5 
any  number  3  times  more  than  1  is,  there-     TT-T-Q -\/  -_•      i~=\  \ 

fore  (U-£=3  times  f—  -^  ,  but  §  is  con- 
tained in  a  number  ^  as  many  times  as  ^, 

therefore,  £-^-f=-5-xt— If  the  same  as  multiplying  f  by  the  divisor 
inverted. 


Ex.  12.—  Divide  6  by  £. 


or  6X1,  the  divisor 


Process.—  6=$,  therefore,  6-^f=f-^f= 
inverted. 

Mixed  numbers  may  be  reduced  to  improper  fractious  and  then 
divided. 

KULE.  —  Multiply  the  dividend  by  the  divisor  inverted. 

Another  method  is  to  reduce  the  fractions  to  a  common  denomina- 
tor and  divide  their  numerators. 

EXAMPLES. 


Divide  — 

(13.)     f    by    f 
(14.)     f    by    f 
(15.)     6    by    | 
(16.)     4fby    | 

(17.)     7    by  21 
(18.)     }iby3f 
(19.)    5fby2f 
(20.)     f    by  A 
(21.)  16    by  f 

(22.)      !  by 
(23.)  100  by 
(24)    f|  by 
(25.)     «by 

(26.)  12  by 
(27.)  *  by 
(28.)  f  by 


(29.)  7|  by  $ 
(30.)  16  by  51 
(31.)  f  off  by* 


(32.)  31  by  |  of  |f 
(33.)  |  of  5J  by  If  of  8 
(34.)  6fby2f 


MENTAL   EXERCISES. 

If  f  of  a  yard  of  silk  cost  &£,  what  is  the  price  per  yard  ? 

of  S£=$A  and  1  yd. 


Process.—  If  f  yd.  cost  S 
will  cost  4  times  ,/4-:=f  f=$ 


±  yd.  will  cost 
Ans. 


If  3  yards  of  silk  cost  $2^,  what  is  the  price  per  yard  ? 
If  4£  yards  of  silk  cost  $3f  what  is  the  price  per  yard  ? 


138  DIVISION  OF  FRACTIONS. 

If  f  of  a  yard  of  silk  cost  $2,  what  is  the  price  per  yard  ? 

At  f  of  a  cent  each,  how  many  pears  can  be  bought  for  12 
cents  ? 

At  $|  a  bushel,  how  many  bushels  of  oats  can  be  bought  for 
$|  ?  $14  ?  #3}  ? 

What  part  of  $1  will  1  quart  of  nuts  cost  if  5  qts.  cost  $£  ? 

What  will  a  pound  of  candy  cost,  if  f  of  a  pound  cost  8§ 
cents  ? 

What  will  a  pound  of  candy  cost  if  4|  Ibs.  cost  $£  ? 

EXAMPLES  FOE   THE   SLATE. 

35.  If  31  yds.  of  silk  cost  $6£,  what  is  the  price  per  yard  ? 

36.  If  3f  Ibs.  of  tea  cost  $2^-,  what  is  the  price  per  pound  ? 

37.  What  will  a  yard  of  muslin  cost  if  26f  yards  cost  |10  ? 

38.  What  costs  a  yard  of  cloth  if  f  of  a  yard  cost  $2f  ? 

39.  If  3  yards  of  cloth  cost  $15f ,  what  is  it  a  yard  ? 

40.  If  3£  yards  of  cloth  cost  $15£:  what  is  it  a  yard  ? 

41.  If  9  quarts  of  cherries  cost  $J,  what  part  of  $1  will  1 
quart  cost  ? 

42.  What  costs  1  bushel  of  corn  if  ^  of  a  bushel  costs  $£  ? 

43.  If  12  £  bushels  of  wheat  cost  $31£,  what  is  the  price  per 
bushel  ? 

44.  At  $lf  a  bushel,  how  many  bushels  of  wheat  can  be 
bought  for  $26  ? 

45.  If  111  ibs.  of  rice  cost  $1|,  what  is  the  price  per  pound  ? 

46.  At  10|  cts.  a  pound  how  many  pounds  of  rice  can  be 
bought  for  87-|  cents  ? 

47.  At  $f  a- pound,  how  much  butter  will  $6  purchase  ? 

48.  If  22  Ibs.  of  butter  cost  $7^,  what  is  the  price  per  pound  ? 

49.  At  16§  cts.  a  pound,  how  many  pounds  of  sugar  can  be 
bought  for  $2. 33^? 

Art,  107,— Reduction  of  Complex  Fractions  to  simple 
fractions  is  properly  included  in  division  of  fractions,  the 
numerators  being  the  same  as  dividends  and  the  denomi- 
nators the  same  as  divisors, 


PROMISCUOUS  EXAMPLES.  139 


EXAMPLES. 

3 

50.   Keduce   JL  to  a  simple  fraction. 


i       3.5      3V2 

-J-H   -.Xj- 

2 

51    Eeduce  1    *  •  3-  ?»•   ±.   I 
7'  li' 

.     8     2 


53.  From  -f  subtract  f  .     From  |  subtract  i 

4i  4  I  H 

54.  Multiply  I  by};  |  by  ||  by  |. 

55.  Divide  |  by  1;  |  by  };|  by  i. 


Promiscuous  Examples  in  Common  Fractions. 

EXEECISE   I. 

1.  Add  f ,  ft,  i  f  ;  2J,  f ,  f  of  |,  6J  ;  i  of  2fc  f ,  3,  4|. 

2.  From  |  subtract  f  ;  9g     1^  ;  60— 45f ;  16|— |  of  7^. 

3.  Multiply  ft  by  4  ;  3|Xf  5  5XA  ;  I  of  6f  X5f  ; 

4.  Divide  14  by  7  ;  8^-f  ;  10 J-^5i  J  I  of  f-h|  of  3^. 


5.  Add  |,  ^,  f,  | ;  §  of  9f,  f,  3,  4£ ;  5^,  ^  $  of  |,  6. 

6.  From  ^  subtract  | ;  7^— ^  ;  |  of  16— 2^4,  ;  120— 96J. 

7.  Mul.  11  by  ^  ;  7><8f. ;  f  of  ^Vf  of  3. 

8.  Div.  12  by  f  ;  ^^4  ;  16f -^  of  f  ;  22f -f-7£. 


9.  Add  f ,  A,  11,  ^  ;  4i,  21,  3f ,  i  of  f  ;  |  of  18^-,  7*,  8. 

10.  From  T%  subtract  f  ;  12—61 ;  f  of  £§— f  of  f . 

11.  Multiply  i^  by  f  ;  6f  X4  ;  14Xf  ;  |-  of  2Xf  of  6f . 

12.  Divide  §  by  | ;  24^-3^ ;  9|^3  ;  f  of 


140  FRACTIONS. 

13.  Add  H,  &,*,#;  6f ,  4,  5f,  f  of  5i ;  f  of  f ,  f  of  12,  J  of  6J. 

14.  From  ^  subtract  f  ;  14f— 4£  ;  f  of  4— £  of  &. 

15.  Multiply  ,&  by  H ;  8f  X6  ;  12Xf  ;  f  of  HXf  of  T\  of  4|. 

16.  Divide  f  by  f4 ;  38-^-9£  ;  24f  ^-6 ;  15£ 4i  of  lOf . 


17.  Add  |,  f,  if,  £  ;  8J,  16§,  10,  ^  ;  f  of  ft,  f  of  6f  ,10f. 

18.  From  Jf  subtract  t\;  20—  Sf  ;  11-L—  6^;  }$—  J  of  f  . 

19.  Multiply  f  by  ^  ;  15X/o  5  tf  X?  ;  $  of  |f  X3J. 

20.  Divide  A  by  f  ;  16-fi  ;  f  ^6  ;  17£^-§  of  13J. 


EXEBCISE  n. 

21.  How  many  yards  £,  |,  |,  and  |  of  a  yard  ? 

22.  From  8|  subtract  2£. 

23.  What  will  11  apples  cost,  at  ^  of  a  cent  each  ? 

24.  How  many  pounds  of  sugar  at  7^  cents  a  pound,  can  be 
bought  for  48|  cents  ? 

25.  At  5§  cents  a  mile  how  much  will  it  cost  to  travel  20 
miles  ? 

26.  At  7f  cts.  a  qt.,  how  many  quarts  of  cherries  can  be 
bought  for  76§  cents  ? 

27.  Five  pieces  of  muslin  contain  the  following  numbers  of 
yards,  18f  ,  15^,  14£,  16|,  and  f  of  16£  ;  how  many  yards  in  all 
the  pieces  ? 

28.  A  piece  of  cloth  measured  22£  yds.,  of  which  f  remain  ; 
how  many  yards  would  be  left  if  5f  yards  more  should  be  cut 
off? 

29.  At  $1£  a  head,  how  many  sheep  can  be  bought  for  $56f  ? 

30.  What  will  12^  yds.  of  ribbon  cost  at  6£  cts.  a  yard  ? 

EXERCISE  ni. 

31.  How  many  are  6£,  §  of  f  ,  7£,  and  3  ? 

32.  From  7^  subtract  f  of  }f 

33.  At  $11  a  bushel  how  much  wheat  can  be  bought  for  $50  ? 

34.  At  $1£  a  bushel,  how  much  cost  72£  bushels  of  wheat  ? 

35.  At  ISf  cts.  a  pound,  how  much  cost  18}  Ibs.  of  butter  ? 

36.  At  22^  cts.  a  pound,  how  many  pounds  of  butter  can  be 
bought  for  $5.37$  ? 


PROMISCUOUS  EXAMPLES.  141 

37.  Four  cheese  weigh  as  follows  :  25|,  26£,  24f,  and  f  of 
254  pounds  ;  how  much  do  all  of  them  weigh  ? 

38.  From  a  piece  of  cloth  which  contained  27f  yards  have 
been  cut  off  at  different  times.  3£,  4£,  5f ,  and  %  yds. ;  how 
many  yards  are  left  ? 

39.  At  621  cts.  a  pound,  how  much  will  7£  Ibs.  of  tea  cost  ? 

40.  At  G2-|  cts.  a  pound,  how  much  tea  can  be  bought  for  $56^? 


41.  How  many  pounds  are  3£,  2^,  and  f  pounds  ? 

42.  From  7^  yards  of  ribbon  2|  yards  have  been  cut  off, 
how  many  are  left  ? 

43.  How  much  will  9|  Ibs.  of  sugar  cost  at  7^  cts.  a  pound  ? 

44.  At  1^  cts.  a  pound  how  much  meal  can  be  bought  for 
13^  cts.? 

45.  At  15  cents  a  yard,  how  much  cost  37f  yards  of  muslin  ? 

46.  At  62^  cts.  a  bushel,  how  many  bushels  of  corn  can  be 
bought  for  $7.811  ? 

47.  Four  pieces  of  calico  measure  as  follows  :  22|,  20£,  21^, 
and  f  of  25  yds. ,  how  many  yards  in  all  of  them  ? 

48.  A  piece  of  cassimere  contained  16f  yards,  of  which  ^  re- 
main ;  if  3^  yards  more  should  be  cut  off,  how  many  would 
be  left  ? 

49.  At  $6f  a  cord,  how  much  will  8f  cords  of  wood  cost  ? 

50.  At  74^  cts.  a  bushel,  how  many  bushels  of  potatoes  can 
be  bought  for  $4.841  ? 


51.  How  many  are  6£,  £  of  f ,  1,  f  of  f ,  2|  ? 

52.  From  3|  yards  of  linen  2^  yards  have  been  but  off,  how 
many  remain  ? 

53.  At  $1^  a  day  what  will  a  man  receive  for  labor  in  26£ 
days? 

54.  At  9|  cts.  a  pound,  how  many  pounds  of  rice  can  be 
bought  for  53f  cts.  ? 

55.  At  $f  a  bushel,  how  much  will  10  bushels  of  corn  cost  ? 

56.  At  $36  an  acre,  how  much  will  $  of  an  acre  of  land  cost  ? 


142  FRACTIONS. 

57.  How  many  yards  f  of  a  yard  wide  will  line  25  yards  of 
cloth  I  yard  wide  ? 

58.  From  a  hogshead  of  molasses  7f  gal.  have  deen  drawn, 
how  many  are  left  ? 

59.  At  5  cents  a  pound,  how  much  will  9$  pounds  of  fish 
cost? 

60.  At  9^  cts.  a  pound,  how  much  cheese  can  be  bought  for 
$1.42$? 

EXEKCISE  VI. 

61.  How  many  are  5£,  £  of  -fa,  £  of  f ,  and  3£? 

62.  From  4£  take  -j^. 

63.  At  3±  cts.  a  yard,  how  much  will  100  yds.  of  tape  cost  ? 

64.  At  f  of  a  dollar  a  yard,  how  many  yards  of  flannel  can 
be  bought  for  |2ii? 

65.  91  cts.  a  yard,  what  will  15£  yds.  of  muslin  cost  ? 

66.  If  7|  pounds  of  tea  cost  $4£&,  what  is  the  price  per 
pound  ? 

67.  Six  pieces  of  ribbon  measure  as  follows  :  9£,  8^,  lOf ,  7£, 
5  yds. ;  how  many  yards  in  all  the  pieces  ? 

68.  From  a  piece  of  linen  containing  11^  yds.  have  been  cut 
off  at  different  times,  f,  £,  1  of  \  yds.,  how  many  are  left  ? 

69.  At  121  cts.  a  dozen,  what  cost  7|  dozen  eggs  ? 

70.  If  13f  bushels  of  apples  cost  $5.17|,  what  is  the  price 
of  a  bushel  ? 

EXEECISE  VH. 

71.  How  many  bushels  are  5£,  3J,  7£  of  -&,  and  f  of  f  bush- 
els ? 

72.  From  7}  take  f  of  f . 

73.  At  16|  cts.  a  yard,  what  cost  lOf  yards  of  calico  ? 

74.  Paid  $2.53£  for  6f  bushels  of  turnips,  what  is  the  price 
per  bushel  ? 

75.  At  7  £  cts.  a  pound,  what  cost  75^  Ibs.  of  sugar  ? 

76.  At  $12f  a  ton,  how  many  tons  of  coal  can  be  bought  for 
$47ff? 

77.  Four  pieces  of  tape  measure  as  follows  :  23  J,  22^,  24£,. 
and  f  of  24§  yds. ;  how  many  yards  in  all  of  them  ? 


PROMISCUOUS    EXAMPLES.  143 

78.  From  a  cheese  weighing  26  pounds  have  been  sold  to 
different  persons  5^,  4§,  3f  ,  and  6  pounds  ;  how  much  is  left  ? 

79.  At  9|  cts.  a  pound,  what  cost  14|  Ibs.  of  codfish? 

80.  Paid  $4.41it  for  18£  yds.  of  muslin  ;  what  was  the  price 
i>c-r  yard  ? 

EXEBCISE  VIII. 

81.  At  $9i  a  barrel,  how  much  cost  7  bbls.  of  flour  ? 

82.  At  $9£  a  barrel,  how  many  barrels  of  flour  can  be  bought 


83.  If  11  barrels  of  flour  cost  $100|,  what  is  the  price  per 
barrel  ? 

84.  How  many  pounds  of  sugar  are  f  ,  ^,  3£,  and  4  Ibs.  ? 

85.  From  13  subtract  9f 

86.  At  5f  cts.  a  pound,  what  cost  76f  Ibs.  of  iron  ? 

87.  At  5f  cts.  a  pound,  how  many  pounds  of  iron  can  be 
bought  for  $455f  ? 

88.  If  72^  Ibs.  of  iron  cost  $4.62-|,  what  is  the  price  per 
pound  ? 

EXEBCISE  IX. 

89.  At  12^  cts.  a  pound,  how  much  lard  can  be  bought  for 
$5.311? 

90.  At  12£  cts.  a  pound,  how  much  will  16§lbs.  of  lard  cost? 

91.  If  18f  Ibs.  of  lard  cost  $2.061  cts.,  what  is  the  price  per 
pound  ? 

92.  A  boy  paid  58^  cts.  for  an  arithmetic,  18f  cts.  for  a  slate, 
9  cents  for  a  sponge,  4£  cts.  for  a  lead  pencil,  and  12^  cts.  for 
paper,  what  did  he  pay  for  all  of  them  ? 

93.  A  girl  bought  a  handkerchief  for  33^  cts.  and  gave  the 
merchant  37^  cents  ;  how  much  change  should  she  have  re- 
ceived ? 

94.  If  one  person  consume  8|  Ibs.  of  beef  in  a  week,  how 
many  persons  would  consume  5874  Ibs.  in  the  same  time  ? 

95.  If  one  person  consume  8§  Ibs.  of  beef  in  a  week,  how 
much  would  a  family  of  9  consume  in  the  same  time  ? 

96.  If  a  family  of  9  persons  consume  73f  Ibs.  of  beef  in  a 
week,  how  many  pounds  will  one  person  consume  in  the  same 
time? 


144  FRACTIONS. 

EXERCISE  X. 

97.  If  7f  Ibs.  of  tea  cost  $9;I7¥,  what  is  the  price  per  pound"7 

98.  At  $1.18£  a  pound,  how  much  will  6|  Ibs.  of  tea  cost  ? 

99.  At  $lf  a  pound,  how  many  pounds  of  tea  can  be  bought 
for  $8£  ? 

100.  Bought  of  the  butcher  on  Monday,  6-^  Ibs.  of  beef, 
on  Tuesday  5^  Ibs,  on  Wednesday  4£ Ibs. ,  on  Thursday  7|  Ibs., 
on  Friday  3^  Ibs. ,  and  on  Saturday  12§  Ibs. ,  how  many  pounds 
during  the  week  ? 

101.  A  piece  of  beef  weighed  11|  Ibs. ;  after  being  divided 
the  smaller  piece  weighed  4^£  Ibs. ;  how  much  did  the  larger 
weigh  ? 

102.  If  a  tub  of  lard  contains  48^-  Ibs.,  how  many  tubs  will 
582£  Ibs.  fill  ? 

103.  If  ll^tubs  of  lard  contain  618f  Ibs.  of  lard,  how  many 
pounds  must  there  be  in  a  tiib  ? 

104.  If  1  tub  will  contain  46^f  Ibs.  of  lard,  how  many  pounds 
will  12  tubs  hold  ? 

EXEECISE  XI. 

105.  At  43f  cts.  a  bushel,  how  many  bushels  of  oats  can  be 
bought  for  for  $25  ? 

106.  At  43f  cts.  a  bushel,  how  much  will  47f  bushels  of  oats 
cost? 

107.  If  7f  bushels  of  oats  cost  $3.22§|,  what  is  the  price 
per  bushel  ? 

108.  A  horse  ate  4£  bushels  one  week  ;  5^  the  second  ;  4|£ 
the  third  ;  5£  the  fourth  ;  how  many  bushels  did  he  eat  in  the 
four  weeks  ? 

109.  A  man  bought  27f  bushels  of  oats  for  his  horse,  and 
fed  him  f  of  9f  bushels  ;  how  many  had  he  left  ? 

110.  If  a  horse  eat  ^  of  a  bushel  of  oats  in  a  day,  in  how 
many  days  will  he  eat  18f  bushels  ? 

111.  If  a  horse  eat  |  of  a  bushel  of  oats  in  a  day,  how  many 
will  he  eat  in  18|  days  ? 

112.  If  a  horse  eat  5f  bushels  of  oats  in  10-|  days,  how  many 
does  he  eat  in  a  day  ? 


PltOMISCUOUS   EXAMPLES.  145 

EXEECISE   XII. 

113.  If  a  meadow  yield  2£  tons  of  hay  on  an  acre,  and  the 
whole  of  it  yield  13^  tons,  how  many  acres  does  it  contain  ? 

114.  If  a  meadow  yields  0.2f  tons  of  hay  on  an  acre,  and 
contains  6f  acres,  how  much  will  the  whole  of  it  yield  ? 

115.  If  a  meadow  containing  7f  acres  yield  15^  tons  of  hay, 
how  much  does  it  yield  on  an  acre  ? 

116.  A  farmer  has  a  load  of  hay  in  bales  weighing  as  fol- 
low :— 14:7\,   156§,   139£,  144-$,,   153£,  146£  Ibs.  ;  what  is  the 
weight  of  the  whole  load  ? 

117.  A  farmer  having  a  load  of  hay  weighing  24$  cwt.,  and 
finding  it  too  heavy  for  his  team,  took  off  5f  cwt. ;  what  was 
then  the  weight  of  his  load  ? 

118.  At  $9f  a  ton,  how  many  tons  of  hay  can  be  bought  for 
$9871 ? 

119.  At  $9£  a  ton,  how  much  will  50  tons  of  hay  cost  ? 

120.  If  86£  tons  of  hay  cost  $S62£,  what  is  the  price  per  ton? 

EXEKCISE   XIH. 

121.  If  |  of  21  Ibs.  of  soap  cost  f  of  $2J,  what  is  the  price 
per  pound  ? 

122.  If  a  pound  of  soap  cost  8f  cts. ,  how  much  will  14  times 
1£  Ibs.  cost  ? 

123.  If  f  of  21|  Ibs.  of  starch  cost  f  of  f  of  3f  dollars,  what 
is  the  price  per  pound  ? 

124.  A  man  having  a  farm  of  675|  acres,  gave  |  of  f  of  it  to 
his  son,  and  f  of  the  remainder  to  his  daughter  ;  how  many 
acres  did  he  give  to  both  of  them  ? 

125.  If  H  of  45  gals,  of  vinegar  cost  $3^,  what  is  the  price 
per  gallon  ? 

126.  If  1  gallon  of  vinegar  cost  31^  cts.,  how  much  cost  11 
times  it  of  a  gallon  ? 

127.  If  a  man's  expenses  are  30  times  £  of  a  dollar  a  month, 
how  much  will  they  be  for  £  of  1  month  ? 

128.  If  a  man's  expenses  are  f  of  30  times  f  of  a  dollar  for  | 
of  a  month,  how  much  are  they  for  a  month  ? 

7 


146 


DECIMAL  FRACTIONS. 


DECIMAL    FRACTIONS. 

Art,  109,— Decimal  Fractions  are  such  as  express  one 
or  more  of  10  equal  parts  of  anything,  or  of  some  mul- 
tiple of  10,  by  itself ;  as,  100,  1000,  etc. 

The  Denominator,  not  usually  written,  is  always  10, 100, 
or  1000,  etc.;  when  written  the  fraction  is  also  common. 

Since  the  value  of  figures  decreases  in  a  tenfold  ratio 
from  left  to  right,  if,  beginning  at  the  left  hand,  we  di- 
vide the  value  of  any  figure  by  10,  we  find  the  value  of 
the  same  figure  in  the  next  place,  as  in  111  or  222,  1(00) 
-4-10=1(0),  2(00)-4-10=2(0)  etc.  For  the  same  reason 
if  we  continue  dividing  by  10  we  shall  find  the  value  of 
the  same  figure  repeated  any  number  of  times  after  the 
whole  number ;  but  it  will  be  less  than  1,  and  therefore 
a  fraction.  To  distinguish  the  whole  number  from  the 
fraction,  a  decimal  point  (.)  is  placed  between  them, 
thus  111.111,  &c.,  222.2222,  &c. 

In  these  numbers  the  value  of  the  first  decimal  figure 
next  to  the  point  is  1-^10=^  ;  2-^10=1%,  and  of  the 
next  1ir-^10=T£ir,  ^j-^10— j^y,  etc.,  but  they  are  usually 
read  as  one  numerator  having  a  common  denominator  the 
same  as  that  of  the  last  decimal  figure  ;  thus  .11  is  read 
as  rife,  .222  as  ^fifr,  etc. 

Artt  110. — NUMERATION  TABLE. 
"Whole  numbers.  Decimals. 


1234567. 


6784. 


DECIMAL  FEACTIONS.  147 

Bead,  1  million,  234  thousand,  567,  and  (decimal)  235  thou- 
sand, 784  inillionths. 

.2  is  read 2  tenths  ^ 

.23  is  read 23  hundredths  ^ 

.  234  is  read 234  thousandths  = 

.2345  is  read 2,345  ten  thousandths      = 

.  23456  is  read...  23,456  hundred  thous.     = 
.234567  is  read.  .234,567  millionths  = 

Art.  110. — RULE  FOR  BEADING  DECIMALS. — Read  as  whole 
numbers  and  add  the  name  or  denominator  of  the  last  decimal 
figure. 

The  denominator  is  always  1,  with  as  many  ciphers  annexed  as 
there  are  decimals  ;  .25=-^.  •025=T§fTJ. 

Prefixing  a  cipher  to  a  decimal  divides  its  value  by  10,  because  it 
multiplies  its  denominator  by  10  without  changing  the  numerator  ; 
as,  .5=^,  but  .05=ythj  5  annexing  a  cipher  does  not.  alter  the  value, 
because  it  is  the  same  as  multiplying  both  the  numerator  and  de- 
nominator of  a  fraction  by  10  ;  as  .5=,^.  •5Q=-f'oi5=~£Q- 

To  distinguish  the  whole  number  from  the  decimal  in  reading,  use 
the  word  decimal  before  the  decimal  expression,  or  when  the  numbers 
are  concrete,  read  them  as  such  (dollars,  yards;  &c.) 

EXAMPLES   TO   BE   BEAD. 

25;  .6;  2.5;  .025;  .3;  .36;  3.6;  .036;  12.5;  125.;  1.25; 
125;  .0125;  136.;  13.6;  1.36;  .136;  00136;  147.;  .147;  1.47; 
14.7;  .0147;  .00147;  2356.;  .2356;  23.56;  2.356;  .235.6; 
.002356;  200. 02;  100.;. 001;  1728.;  1.728;  172.8;  .1728;  17.28; 
.001728;  2500.25;  .07;  .067;  4.37;  21.21;  300.03;  40.4; 
4.04;  .404;  .000404;  .0005;  31.0031;  .310031;  3.10031. 

Art.  111. — RULE   FOR  WRITING   DECIMALS. —  Write  what 
would  be  the  numerator  of  a  common  fraction  as  a  whole 
number,  and  prefix,  if  necessary,  ciphers  till  the  right  hand 
figure  is  in  its  proper  place,  and  then  the  decimal  point. 

EXAMPLES   TO   BE   WRITTEN. 

Fifteen  ;  fifteen  hundredths  ;  fifteen  thousandths  ;  15  mil- 
lionths ;  (15.;  .15;  015;  .000015;)  5  tenths  ;  5  thousandths  ; 
5  millionths  ;  2  and  5  tenths ;  25  thousandths  ;  25  hundred 


148  DECIMAL  FRACTIONS. 

thousandths ;  333  thousandths  ;  33  thousandths  ;  3  thousandths; 
27  and  27  thousandths ;  3  and  45  hundred  thousandths  ;  36 
hundredths ;  3  tenths  and  6  hundredths ;  356  thousandths ;  3 
tenths,  5  hundredths  and  6  thousandths  ;  28  hundredths ;  128 
ten  thousandths  ;  8  millionths  ;  7  hundred  thousandths  ;  4  and 
4  tenths ;  200  and  2  hundredths ;  3000  and  3  thousandths ; 
45  millionths  ;  175  ten  thousandths. 
(See  examples  in  Addition,  &c.) 

Art*    112* — APPLICATION  OF  THE   FUNDAMENTAL,   RULES  TO 
DECIMALS. 

Since  decimals  increase  from  right  to  left  in  a  tenfold 
ratio,  the  same  as  whole  numbers,  to  which  they  are  com- 
monly annexed,  they  may  be  added,  subtracted,  multi- 
plied, and  divided  by  the  same  rules  except  in  a  few  par- 
ticulars. 


Art.  113,— Addition  of  Decimals, 

EXAMPLE  1.— Add  6  and  5  tenths ;  10  and  1  hundredth  ;  250 
and  25  thousands  ;  144  and  265  thousandths  ;  14  and  4  tenths  ; 
144  thousandths. 

Process.—  6.5 

10.01 
250.025 
144.265 
14.4 
.144 

Ans.    425.344" 

RULE. —  Write  the  numbers  so  that  all  the  decimal  points 
will  be  under  one  another ;  atid  as  in  whole  numbers  and 
place  the  decimal  point  in  the  sum  under  the  others. 

Pupils  are  supposed  to  be  familiar  with  addition,  and  the  follow- 
ing examples  are  chiefly  designed  to  exercise  them  in  writing  deci- 
mals. If  they  find  the  correct  answers  they  will  probably  have  writ- 
ten the  numbers  correctly. 


ADDITION   OF  DECIMALS.  149 

EXAMPLES. 

2.  Add  2  and  17  hundredths  ;   13  and  6  thousandths  ;  12  and 
145    thousandths ;   10  and  93    thousandths ;   17  and    81    ten 
thousandths ;  75  and  708  hundred  thousandths ;  16  and  456 
thousandths. 

3.  Add  26  and  5001  ten  thousandths;  37  and  604  thousandths; 
8  and  77  hundredths  ;  15  and  708  thousandths;  98  and  7  tenths ; 
1.99  hundredths  ;  18  and  45  thousandths. 

4.  Add  61  and  4  thousandths  ;  4  and  7  hundredths  ;  329  and 
8  tenths  ;  47  and  39  hundredths  ;  731  and  96  thousandths  ;  5 
and  5  ten  thousandths  ;  6  and  8  tenths. 

5.  Add  42  and  8  hundredths; ;  521  and  28  thousandths  ;  63 
and  125  ten  thousandths ;  108  and  215  thousandths ;  14  and 
25  hundredths  ;  9  and  5  tenths ;  18  and  23  hundredths ;  110 
and  11  thousandths. 

6.  Add  17  and  55  thousandths ;  4  and  81  hundredths  ;  90  and 
1935  ten  thousandths  ;  77  and  85  hundred  thousandths  ;  24 
and  106  ten  thousandths  ;  35  and  7  tenths. 

7.  Add  12  and  5  tenths  ;  13  and  65  hundredths  ;  114  and  25 
thousandths  ;   46  and  121  ten  thousandths  ;   64  and  7  tenths  ; 
127  and  18  thousandths  ;    43  and  9  hundredths  ;    102  and  6 
tenths. 

8.  Add  6  and  157  thousandths ;  18  and  225  ten  thousandths ; 
172  and  16  hundredths;  27  and  81  thousandths;   9  and  23 
hundred  thousandths  ;  13  and  27  thousandths ;  6  and  12  hun- 
dredths. 

9.  Add  12  and  9  thousandths ;  125  and  8  tenths ;  245  ten 
thousandths  ;  249  ;  63  and  63  hundredths. 

10.  Add  17 ;  17  hundreds  ;  17  thousandths ;  17  hundredths ; 
1  and  7  tenths. 

11.  Add  16  thousand ;  16  hundreds ;  160;  16 ;  1  and  6  tenths; 
16  hundredths  ;  16  thousandths. 

12.  What  is  the  sum  of  8  and  75  hundredths;  60  and  7 
tenths  ;   12  and  5  thousandths ;  180  and  27  hundredths  ;   29 
and  21  thousandths  ;  3  and  15  hundredths  ? 

13.  What  is  the  sum  of  3  thousand  and  6  hundredths ;  2 


150  DECIMAL  FRACTIONS. 

hundred  and  45  thousandths  ;  10  and  1  hundredth  ;  4  thousand 
and  6  hundredths  ;  4  hundred  and  6  thousandths  ? 

14.  What  is  the  sum  of  10  and  1  tenth ;   100  and  1  hun- 
dredth ;  1000  and  1  thousandth ;  200  and  2  thousandths ;  20 
and  2  hundredths  ? 

15.  What  is  the  sum  of  25  tenths ;  126  hundredths ;  1354 
thousandths  ;   13579  ten  thousandths  ;   2468  thousandths  ;  357 
hundredths  ? 


Art.  114,    Subtraction  of  Decimals. 

EXAMPLE  1. — From  123  and  56  hundredths,  subtract  12  and 
156  ten  thousandths. 

Process.—    123.56 

12.0156 
Ans.     111.4444 

RULE. —  Write  the  numbers  so  that  the  decimal  points  will 
be  under  each  other,  subtract  as  in  whole  numbers,  and  place 
the  decimal  point  in  the  remainder  under  the  others. 

If  there  is  no  figure  directly  above  the  one  to  be  subtracted,  con- 
sider the  place  as  filled  with  a  cipher. 

EXAMPLES. 

2.  From  1.  subtract  1  tenth. 

3.  From  20.  subtract  2  hundredths. 

4.  From  250.  subtract  25  thousandths. 

5.  From  1356.  subtract  356  ten  thousandth. 

6.  From  23464.  subtract  3464  hundred  thousandths. 

7.  From  100,000.  subtract  100  thousandths. 

8.  From  1,000,000.  subtract  1  millionth. 

9.  From  8  tenths  subtract  4  hundredths. 

10.  From  75  hundredths  subtract  75  thousandths. 

11.  From  56  thousandths  subtract  56  ten  thousandths. 

12.  From  5  tenths  subtract  5  thousandths. 

13.  From  5  hundredths  subtract  5  hundred  thousandths. 

14.  From  5  thousandths  subtract  5  millionths. 

15.  From  5  subtract  3  tenths. 


MULTIPLICATION   OF  DECIMALS.  151 


"What  is  the  difference  between — 

16.  10.  and  .01  ? 

17.  5.  and  .5  ? 

18.  600  and  6  hundredths  ? 

19.  7000  and  7  thousandths  ? 

20.  80000  and  8  ten  thousandths  ? 

21.  3  and  .3  ? 

22.  3  tens  and  3  tenths  ? 

23.  3  hundreds  and  3  hundredths  ? 

24.  3  thousands  and  3  thousandths  ? 

25.  3  millions  and  3  millionths  ? 


Art.  115.— Multiplication  of  Decimals. 

General  Principle. — The  denominator  (understood)  of 
any  product  of  decimals  is  the  product  of  their  denom- 
inators ;  thus  the  product  of  tenths  and  tenths  is  hun- 
dredths (iVXiV^Tihy);  the  product  of  tenths  and  hun- 
dredths is  thousandths,  (i1ijXTou=roW»)  &c.t  &c. 

EXAMPLE  1.— Multiply  1.25  by  .125. 

Process — The  same  as  in  whole  numbers,  except  1.25 

five  figures  are  pointed  off  in  the  product  for  deci-  .125 

mals,  because  the  product  of  hundredths  and  thous-  g25 

andths  is  hundred  thousandths.     1. 25=l-,s-(f j=|^f  ;  OKA 

.125=-^%  ;  and  I^XiV&^iWo^^.  15625,  Ana.  125 


Ans.  .15625 

BULE. — Multiply  as  in  whole  numbers,  and  point  off  from 
the  right  of  the  product  as  many  figures  for  decimals  as  there 
are  decimals  in  both  the  multiplicand  and  multiplier ;  or  so 
that  the  denominator  (understood)  of  the  product  of  the  de- 
cimals shall  be  the  product  of  their  denominators. 

If  there  be  not  figures  enough  in  the  product,  prefix  ciphers. 

Ex.  2.— Multiply  .256  by  100. 

Process.— .256X100=25.600  or  simply  remove  the  decimal  point 
two  places  to  the  right. 


152 


MULTIPLICATION   OF    DECIMALS. 


SPECIAL  RULE. — To  multiply  by  10,  100,  &c.:  remove  the 
decimal  point  as  many  places  to  the  right  as  there  are  ciphers 
in  the  multiplier,  annexing  ciphers  if  necessary. 

EXAMPLES. 

[Let  pupils  write  the  following  examples  with  the  respective  pro- 
ducts on  their  slates  ;  also  recite  the  answers  without  having  them 
written,  till  they  can  do  so  readily  and  without  mistakes.] 


Multiplies 
Multipliei 
Products 

mds  . 

'S.  .  . 

.  3        3 
2        2 

.3 

.2 
.06 

.06 
.05 

.03 
.2 

.006 

7 
6 

.03 

.02 

.3 
.3 

.03 
.3 
J009 

.07 
.06 

6      76 

.0006 

7        .7 
.6        .6 

.09 

.07 
.6 

6           6 
5          .5 

.6 
.5 

.06 
.5 

8 

7 

10 
9 


8 
.7 

10 
.9 


.7 
1.0 


.08 


.10 
.9 


.08 

.07 


.10      11       11       11 
.09      10      1.0      .10 


.09 


1.1 

1.0 


.08 

.11 
.10 


12 
11 


12 
1.1 


12 
.11 


.12 
1.1 


.12 
.11 


12 
12 


12       12 
1.2      .12 


.12 
1.2 


.12 
.12 


3 

2  tenths. 


4  tenths.     5  hund'ths.     6  hund'ths.     7  hund'ths. 
3  tenths.     4  tenths.         5  hund'ths.     6  thous'ths. 


4  5  tenths.     6  hund'ths.     7  hund'ths.     8  hund'ths. 

3  tenths.     4  tenths.     5  tenths.         6  hund'ths.     7  thous'ths. 


5 

4  tenths. 

6 

5  tenths. 


6  tenths.     7  hund'ths.     8  hund'ths.     9  hund'ths. 
5  tenths.     6  tenths.         7  hund'ths.     8  thous'ths. 


7  tenths. 
6  tenths. 


8  hund'ths. 
7  tenths. 


9  hund'ths.  10  hund'ths. 
8  hund'ths.     9  thous'ths. 


12  thousandths. 
11  tenths. 


12  ten  thousandths. 
11  hundredths. 


12  millionths. 
11  thousandths. 


EXAMPLES  FOE  THE   SLATE,    ETC. 

3.  Multiply  21  and  6  tenths  by  3  and  6  hundredths. 

4.  156  and  25  thousandths  by  2  and  75  hundredths. 
6.  50  and  5  hundredths  by  2  and  16  thousandths. 


DIVISION  OF  DECIMALS.  153 

6.  175  thousandths  by  100. 

7.  22  ten  thousandths  by  11  hundredths. 

8.  18  by  256  thousandths. 

9.  6  and  5  tenths  by  65  hundredths. 

10.  325  thousandths  by  50. 

11.  672  ten  thousandths  by  25. 

12.  1  millionth  by  1000. 

13.  100  by  1  thousandth. 

14.  5  thousandths  by  4  thousandths. 

15.  125  milHonths  by  1,000,000. 

16.  275  and  275  thousandths  by  25  and  25  hundredths. 

In  U.  S.  Money  cents  and  mills  are  decimal  fractions  of  a  dollar  ; 
1  cent=$.01 ;  25  cents=$.2o  ;  50  cents=$.5  ;  1  mill=:$.001  ;  5 
mills=$.005,  &c. 

17.  At  12  cts.  a  yard,  how  much  will  16. 5  yds  of  calico  cost  ? 

18.  At  75  cts.  a  bushel,  how  much  will  18. 25  bushels  of  com 
cost? 

19.  At  62  cts.  5  m.  a  gallon,  how  much  will  20.5  gals,  of  mo- 
lasses cost  ? 

20.  At  $8.625  a  ton,  what  cost  6.5  tons  of  coal  ? 

21.  At  $100  an  acre,  what  cost  63.75  acres  of  land  ? 

22.  At  $9.625  a  barrel,  what  cost  20  barrels  of  flour  ? 

23.  At  $0.1875  a  pound,  what  cost  37.5  Ibs.  of  lard  ? 

24.  At  $.025  a  mile,  how  much  will  it  cost  to  travel  100 
miles? 


Art.  116.— Division  of  Decimals, 

General  Principle.  The  denominator  (understood) 
of  the  dividend  divided  by  that  of  the  divisor  is  the  de- 
nominator of  the  quotient ;  thus  thousandths  divided  by 
tenths  are  hundredths,  and  hundredths  divided  by  tenths 
are  tenths  dolo  '.  IO=TOU  >  i&o  '  i1o=Ar)  &c. 

EXAMPLE  1.— Divide  15.625  by  .25. 

7* 


154  DIVISION   OF  DECIMALS. 

Process. — The  same  as  in  whole  numbers,  ex-  ,25)15.625(62.5 
cept  one  figure  is  pointed  off  in  the  quotient  be-  15  0          Ans. 

cause  thousandths  divided  by  hundredths,   are  ^3 

tenths ;  50 

15625  25         ,1  12^ 

and 

625_.?0 

'- 

EULE. — Divide  as  in  whole  numbers,  and  from  the  right  of 
the  quotient,  point  off  as  many  figures  for  decimals  as  the  de- 
cimals in  the  dividend  exceed  those  in  the  divisor  ;  or,  so  that 
the  denominator  (understood)  in  the  quotient,  shall  be  the  quo- 
tient of  the  denominator  in  the  dividend  divided  by  that  of 
the  divisor. 

If  there  are  not  figures  enough  in  the  quotient,  prefix  ciphers 
to  the  decimals,  or  annex  them  to  whole  numbers. 

If  there  are  not  as  many  decimals  in  the  dividend  as  in  the  divisor, 
annex  ciphers.  If  other  ciphers  are  annexed  to  the  remainder,  they 
must  be  considered  as  filling  decimal  places  in  the  dividend. 

Ex.  2.— Divide  3.25  by  1-0. 

Process. — Remove  the  point  two  places  to  the  left,     Ans.  .0325 
prefixing  a  cipher  where  a  figure  is  wanting,  which  is 
the  same  as  dividing  by  100. 

SPECIAL  EULE. — To  divide  decimals  by  10,  100,  &c.,  re- 
move the  decimal  point  as  many  places  to  the  left  as  there  are 
ciphers  in  the  divisor,  prefixing  ciphers  if  necessary. 
EXAMPLES. 

[Let  the  pupils  write  the  following  examples  with  the  respective 
quotients,  on  their  slates  ;  also  recite  the  answers  without  seeing 
them  till  they  can  do  so  readily  and  without  mistakes.] 

2)6    .2)6  .2).06     .2). 006  .02). 0006    .3). 09     .3). 009    .03). 0009 
4)24     .4)2.4     .4).24     .4). 024.     .04). 024.    .004).Q24 
5)30        .5)30          .05)30         6)42         .6)42        .6^4.2 
7)5_6       .7)56          .7)5.6        .07).  56      .8)72        .8). 72 
9)90       .9)90         .09)_.90_        8).96      .8)9.6     .08).96 


DIVISION  OF  DECIMALS.  155 

11)132         .11)132        11)13.2        1.1)132         .11)1.32 
12)144         .12)144        12)14.4       1.2)144        .12)1.44 
Divide — 


6  tenths  by  2  tenths. 

6  hundredths  by  2  tenths. 

6  thousandths  by  2  tenths. 

9  (whole  number)  by  3  hun'hs 

9  hundredths  by  3  hund'ths. 

9  thousandths  by  3  tenths. 
12  thousands  by  4  hund'ths. 
12  hundredths  by  4  tenths. 


16  tenths  by  4  tenths. 

16  thousandths  by  4  hund'ths. 

24  hundredths  by  4  hund'ths. 

32  tenths  by  8  tenths. 

36  thousandths  by  9  hund'ths. 

45  hundredths  by  5  tenths. 

56  millionths  by  7  thous'ndths. 

64  millionths  by  8  hundredths. 


EXAMPLES  FOE  THE   SLATE. 

3.  Divide  1728  by  12,  .12,     1.2,  .012 

4.  Divide  .1728  by  12,  .12,     1.2,  .012 

5.  Divide  17.28  by  12,  .12,     1.2,  .012 

6.  Divide  1.728  by  12,  .12,     1.2,  .012 

7.  Divide  172. 8  by  12,  .12,     1.2,  .012 

8.  Divide  1728  by  144,  .144,  1.44,  14.4 

9.  Divide  17.28  by  144,  .144,  1.44,  14.4 

10.  Divide  1.728  by    144,  .144,  1.44,     14.4 

11.  Divide  172.8  by    144,  .144,  1.44,     14.4 
Divide — 

12.  34  and  11  hundredths  by  9  and  8  tenths. 

13.  125  and  18  thousandths  by  5  and  25  hundredths. 

14.  40  and  215  ten  thousandths  by  8  and  5  tenths. 

15.  25  ten  thousandths  by  25  hundredths. 

16.  1  thousandth  by  1000. 

17.  63  and  63  hundredths  by  7  tenths. 

18.  125  millionths  by  500. 

19.  At  $5. 75  a  cord,  how  many  cords  of  wood  can  be  bought 
for  $50  ? 

20.  If  7.5  cords  of  wood  cost  $38.775,  what  is  the  price  of  a 
cord? 

21.  If  75.97  acres  of  land  cost  $2696.935,  what  is  the  price 
of  an  acre  ? 


156  DECIMAL  FRACTIONS. 

22.  At  $35.50  an  acre,  how  many  acres  of  land  can  be  bought 
for  $1348. 4625. 

23.  If  12.75  yards  of  cloth  cost  $97.5375,  what  is  the  price 
per  yard  ? 

24.  At  7. 65  a  yard,  how  many  yards  of  cloth  can  be  bought 
for  $91.80  ? 

25.  At  $7.375  a  ton,  how  many  tons  of  coal  can  be  bought 
for  $176. 63125? 

26.  If  100  tons  of  coal  cost  $737.50,  what  is  the  price  per 
ton? 


Art,  117,—  Promiscuous  Examples  in  Decimal  Fractions, 
including  U,  S,  Money. 

EXEECISE   I. 

1.  Bought  tea  for  1  dollar  25  cts.  ;  sugar,  3  dollars  37  cts. 
5  m.  ;  starch,  50  cts.  ;  molasses,  2  dol.  9  cts.  ;  ginger,  18  cts.  5 
m.     What  is  the  amount  of  the  bill  ? 

2.  A  man  having  $10,000,  paid  $6,521.875  for  a  farm  ;  how 
much  had  he  left  ? 

3.  At  $1.875  a  bushel,  how  much  will  12.5  bushels  of  wheat 
cost? 

4.  If  16.25  bushels  of  wheat  cost  $28.4375,  what  is  the  price 
per  bushel  ? 

5.  At  $1.75  a  bushel,  how  many  bushels  of  wheat  can  be 
bought  for  $30  ? 

6.  How  many  hundred  weight  of  sugar  in  five  barrels  weigh- 
ing 2  cwt.  75  hundredths  ;  2  cwt.  875  thousandths  ;  3  cwt.  4 
hundredths  ;  2  cwt.  5  tenths  ;  3  cwt.  1275  ten  thousandths  ? 

7.  A  merchant  having  a  barrel  of  sugar  weighing  2  cwt.  has 
sold  from  it  1  cwt.  375  thousandths  ;  how  much  of  it  is  left  ? 

8.  If  2.5  yards  of  cloth  will  make  a  coat,  how  many  yards 
will  make  15  coats  ? 

9.  If  2  375  yards  of  cloth  will  make  a  coat,  how  many  coats 
will  28.5  yards  make  ? 

10.  If  37.5  yds.  of  cloth  will  make  15  coats,  how  many  yards 
will  make  one  coat  ? 


PROMISCUOUS  EXAMPLES.  157 

EXERCISE  H. 

11.  Bought  a  barrel  of  iiour  for  10  dol.  50  cts.  ;  a  bushel  of 
timothy  seed  for  4  dol.  61;  cts.  5  m.  ;  a  cheese  for  3  dol.  6  cts. 
5  m. ;  a  box  of  soap  for  8  dollars,  and  a  broom  for  38  cts.  ;  what 
is  the  amount  of  the  bill  ? 

12.  A  lady  having  purchased  a  bill  of  goods  amounting  to  4 
dollars  6  cts. ,  gave  the  clerk  a  10  dollar  bill ;  how  much  change 
was  due  her  ?. 

13.  At  $1.37  a  pound,  how  much  will  3.25  pounds  of  tea 
cost  ? 

14.  At  $1.375   a  pound,  how  many  pounds  of  tea  can  be 
bought  for  $5  ? 

15.  If  4.75  pounds  of  tea  cost  $6.00,  what  is  the  price  per 
pound  ? 

16.  How  many  hundred  weight  of  meal  in  4  bags  weighing 
1  cwt.  1875  ten  thousandths  ;  1  cwt.  57  hundredths ;  1  cwt.  ;  .98 
cwt.  ? 

17.  A  miller  ground  6  cwt.  of  meal  and  sold  3  cwt.  125  thou- 
sandths ;  how  much  of  it  was  left  ? 

18.  If  a  barrel  contain  2.75  bushels  of  potatoes,  how  many 
bushels  will  there  be  in  12.5  barrels  ? 

19.  If  a  barrel  will  contain  2.5  bushels  of  potatoes,  how 
many  barrels  will  contain  10.75  bushels  ? 

20.  If  100  bushels  of  potatoes  be  put  in  36.5  barrels,  how 
many  must  each  barrel  contain  ? 

EXEECISE  HI. 

21.  Bought  a  load  of  potatoes  for  $15.75  ;  of  turnips  for 
$7.375  ;  of  carrots  for  9  dollars,  8  cts.  5m.;  of  beets  for  $10 ; 
to  what  did  they  all  amount  ? 

22.  Exchanged   a    horse  worth  $225,  for  a  pair   of  oxen 
worth  $180.625,  and  the  rest  in  cash  ;  what  was  the  amount  of 
cash? 

23.  At  $6.375  a  yard,  how  much  will  3.75  yds.  of  cloth  cost? 

24.  At  $6.625  a  yard,  how  much  cloth  can  be  bought  for 
$25? 

25.  If  3.8  yds.  of  cloth  cost  $20.5,  what  is  the  price  per 
yard? 


158  DECIMAL  FKACTIONS. 

26.  How  many  acres  of  land  in  five  fields  measuring  4  A.  27 
hundredths  ;  6  A.  28  thousandths  ;  5  A. ;  7  A.  7  hundredths  ; 
6  A.  5  tenths  ? 

27.  If  a  field  measures  10  A.  5  hundredths,  and  5  A.  5  tenths 
be  fenced  off  from  it,  how  much  of  it  will  be  left  ? 

28.  If  a  field  contains  6.54  acres,  how  many  acres  will  there 
be  in  10  fields  of  the  same  size  ? 

29.  If  8  fields  of  equal  size  contain  45.75  acres,  how  many 
acres  in  each  ? 

30.  If  a  field  contain  6.45  acres,  how  many  such  fields  will 
contain  58.05  acres  ? 

EXEKCISE  IV. 

31.  Bought  a  carpet  for  $12  ;  matting,  $6 ;  carpet  binding, 
50  cts.  ;  tacks,  6  cts.  ;  what  was  the  amount  of  the  bill  ? 

32.  A  clerk's  salary  is  $1,000  a  year,  and  his  expenses  $656.- 
625.  how  much  can  he  lay  up  ? 

33.  If  a  clerk  receives  $1,000  a  year  (313  working  days),  how 
much  is  it  a  day  ? 

34.  If  a  man  earn  $2.626  a  day,  how  much  will  he  earn  in 
300.5  days  ? 

35.  If  a  man  earn  $572.5  in  312.75  days,  how  much  does  he 
earn  in  a  day  ? 

36.  How  many  pounds  in  4  hams  weighing  20  and  2  tenths 
pounds ;  21  and  28  hundredths  ;  19  Ibs. ;  18  and  25  thousandths? 

37.  A  ham  weighed  22.5  Ibs. ;  after  being  smoked  it  weighed 
19  Ibs.  8125  ten  thousandths  ;  how  much  weight  had  it  lost  ? 

38.  If  each  firkin  of  butter  contains  54.75  Ibs.,  how  many 
pounds  will  there  be  in  10. 5  firkins  ? 

39.  If  a  firkin  will  hold  54.24  Ibs.  of  butter,  how  many  fir- 
kins will  hold  1000  Ibs  ? 

40.  If  512. 25  pounds  of  butter  be  packed  in  9 .75  firkins,  how 
many  pounds  will  there  be^in  each  ? 

EXERCISE  v. 

41.  If  a  man's  income  is  $5,000  a  year,  and  his  expenses 
$10'a  day,  how  much  will  he  hav.e  left  ? 

42.  If  a  man's  income  is  $5,000  a  year  (365  days),  how  much 
may  he  expend,  and  lay  up  $5.00  every  day  ? 


REDUCTION   OF   COMMON  FK ACTIONS.  159 

43.  If  a  man  lay  up  $2.000  a  year  (or  313  working  days), 
how  much  will  he  lay  up  each  day  on  an  average  ? 

44.  How  much  butter  in  four  tubs,  weighing  16.5  Ibs. ;  20 
Ibs. ;  18.25  Ibs. ;  19  and  25  thousandths  pounds  ? 

45.  If  each  load  of  coal  weigh  18. 75  cwt. ,  how  much  will  20 
loads  weigh  ? 

46.  If  a  man  draw  with  his  team  20.5  cwt.  of  coal  in  24  loads, 
how  much  must  he  draw  at  a  load  ? 

47.  If  a  man  draw  16. 75  cwt.  at  a  load,  how  many  loads  will 
251.25  cwt.  make  ? 

48.  If  a  man  chop  3.25  cords  of  wood  in  a  day,  how  long  will 
it  take  him  to  chop  30  cords  ? 

Art*  118. — REDUCTION  OF  COMMON  FRACTIONS  TO  DECI- 
MALS. 

EXAMPLE  1. — Eeduce  f  to  a  decimal. 

Process. — Since  ^=3-f-4  and  4  is  not  contained  in  3  a 
whole  number  of  times,  we  find  the  value  of  3  in  the  first 
place  of  decimals  to  be  3.0  (30  tenths)  the  same  as  the         4)3.00 
numerator,  with  a  cipher  annexed,  which  divided  by  4,    ^ns'    "75 
is  .7-)-. 2  remainder,  but  .2=. 20  the    same  as  annexing 
another  cipher  to  the  numerator,  and  .20-^-4=.  05,  which 
added  to  .  7=.  75,  Ans. 

RULE. — Add  ciphers  to  the  numerator  and  divide  by  the 
denominator. 

Ex.  2.  Reduce  ^  to  a  decimal. 
Protess.—  7)1.000000      or     7)1.000 
"142857+  Tl43 

In  this  Arithmetic,  whenever  a  decimal,  as  in  the  above  example, 
will  extend  to  more  than  four  places,  three  places  will  be  considered 
sufficiently  accurate  if  it  is  to  be  used  afterwards,  but  if  the  figure  in 
the  fourth  place  would  be  more  than  five,  1  will  be  added  to  the 
figure  in  the  third  or  thousandths  place  ;  as  .143  for  .14283. 

Ex.  3.  Reduce  £  to  a  decimal. 
Process.— 3)LOOOOO()    or    3)1.0 
.333333  3 

2.  A  decimal,  consisting  of  the  same  figure  repeated  (as  in  the 
above  example),  or  several  figures,  is  called  a  repetend  or  circulating 
decimal,  and  is  distinguished  as  such  by  a  (.)  dot  over  the  first  and 


160  DECIMAL  FRACTIONS. 

last  figures  repeated.     If  only  a  part  of  the  decimal  is  repeated  it  is 
called  a  mixed  repetend. 

Art,   119, — KEDUCTION    OF    DECIMAL  TO  COMMON    FRAC- 
TIONS. 

EXAMPLE  1.— Eeduce  .375  to  a  common  fraction. 
Process.—  .375=^,  which  reduced  to  its  low-      25)375    515    3 
ost  terms  is  f.  1000     15     8 

KULE. — Erase  the  decimal  point,  and  write,  for  the  denom- 
inator of  the  common  fraction,  1  with  as  many  ciphers  an- 
nexed as  there  are  are  decimal  figures  ;  then  reduce  the  frac- 
tion to  its  lowest  terms. 

Ex.  2.— Eeduce  .3  or  333333,  &o.,  to  a  common  fraction 

Process.— Since  .3=.3^  (£  reduced  to  a  decimal)  and  3  .  3  I 
or  f ,  is  &  less  than  3£  (or  *£\  .3  is  -fj  less  than  .3  ;  but  if  3^9~3 
the  denominator  of  -,%  be  diminished  in  the  same  proportion  the  value 
of  the  fraction  is  not  altered.  Therefore,  3  or  -$j=$,  which  equals  i. 

If  the  repetend  consist  of  more  than  one  figure  the  denominator 
of  the  common  fraction  will  be  as  many  nines  as  there  are  figures 
repeated  ;  thus,  .123=^=^. 

"When  the  decimal  is  a  repetend  write  for  the  denominator  of 
the  common  fraction  as  many  9's  as  there  are  figures  in  the  repetend. 

Ex.  3.  Eeduce  .16  to  a  common  fraction. 

When  only  a  part  of  the  decimal  is  repeated  it  is  called  a  mixed 
decimal,  and  is  the  same  as  a  mixed  number  in  a  complex  fraction  ; 
thus  : 

.lfe=.lf==_li=$ft=k  Ans. 
10 

EXAMPLES. 

Reduce  the  common  fractions  to  decimals,  and  the 
decimals  to  common  fractions. 


CM  *;      f;  I;      I 

(2.)  .5;     .25;  .75;  .125 

(3.)  f;       1;  f;     A 

(4.)  .375;  .625;  .875;   .025 


(50  «;  H;  «;  M 

(6.)  .8;  .16;  .075;  .225 

(7.)  £;  &;  i;  £§ 

(8.)  .6;  027;  .123;  .16 


FRACTIONAL  COMPOUND   NUMBERS.  161 

Fractional  Compound  Numbers. 

Art.  120. — Fractional  compound  numbers  are  com- 
pound numbers  in  the  form  of  fractions  ;  as  jB| ;  j  cwt. ; 
I  gal. 

GENERAL  RULE. — Proceed,  as  in  similar  cases,  in  com- 
pound whole  numbers,  using  the  rules  for  fractions  when 
necessary  or  convenient. 

Special  rules  will  also  be  given  in  some  cases,  but  it  is  better  to 
understand  and  apply  the  general  rule. 

CASE  I. 

Art.  121. —  Compound  numbers  reduced  to  fractional  com- 
pound numbers  ;  reduction  ascending. 
EXAMPLE  1. — Eeduce  10s.  6d.  to  the  fraction  of  a  pound. 

Process. — By  common  fractions.  6d.         12)6  d. 
—  12  =  -A-s.  =is.,    to  which    add    or    OOMO&  «      «-    omiftl 
prefix  the  10s.2   the  sum  10is.^20=    '®£tt  ^^ 

£$k-     This  common  fraction  reduced    ^240— tft  40 — BW 

to  a  decimal  is  £.525. 

Process.— By  decimal  fractions,  6d.-^-12=  12)6.0 

6.0d.-^12=.5s.,  to  which  add  or  prefix  the       20)10~5 
10s.     'The  sum  10.5s.-:-20  =  £.525  =  £V$fc  '  ajL_2 

• — "ftoo — \ 


SPECIAL  RULE. — Beginning  with  the  least  denomination 
given,  reduce  it  to  the  next  greater  by  division  of  fractions 
(common  or  decimal),  and  add,  or  annex  it  to  any  given 
number  of  the  same  denomination  or  name.  Proceed  thus  till 
the  required  fraction  is  found,  which  reduce  to  its  lowest 
terms. 

For  convenience  write  the  least  denomination  first,  and  the  others 
under  it  in  order. 

EXAMPLES. 

[Let  the  answers  be  found  in  both  common  and  decimal 
fractions.] 
Ex.  2.  What  part  of  a  cwt.  is  2  qr.  10  Ibs.  ? 

3.  What  part  of  a  mile  is  26  rods  11  ft.  ? 

4.  What  part  of  a  hhd.  is  15  gal.  3  qts.  ? 


162  FRACTIONAL  COMPOUND   NUMBERS. 

5.  What  part  of  a  yard  is  3  qr.  3  na.  ? 

6.  What  part  of  a  day  is  6  h.  30  m.  ? 

7.  What  part  of  an  acre  is  2  B.  10  rods  ? 

8.  What  part  of  a  yard  is  3  qr.  3|  na.  ? 

Process.—  3|  na.  -f-  4=V-  na.  -|-  4=1!  qr.     3H  qr.-r*=tt  yds. 

9.  What  part  of  £3  is  15  s.  6  d.  ? 

Process. — First  find  what  part  of  £1,  and  then  £  of  that. 

CASE  II. 

Art*  122. — Fractional  compound  numbers  reduced  to  in- 
tegral compound  numbers  ;  reduction  descending. 
EXAMPLE  1 . — Eeduce  £f  to  whole  numbers. 

Process. — By  common  fractions.  Since  £y  is  less  than  £1,  reduce 
it  to  shillings.  £$  X  20  =  ^s. = 14$s. ;  f  s.  X12  =  yd. = Sfd. ;  f  d. 
X4=V-fer.  r=  If  far.  Therefore,  £$=14s.  3d.  l^far.,  Ans. 

Ex.  2.  Keduce  £.625  to  whole  numbers. 

-  .625£. 


Process  (by  decimal  fractions'),  the  same  as  in  reduc- 


20 


tion  descending  of  whole  numbers,  except  pointing  off,       12.500s. 
ns  in  multiplication  of  decimals.  12 


Ans.  12s   6.006d. 

SPECIAL  RULE. — Eeduce  the  fractions  to  kss  denomina- 
tions, and  find  the  value  of  each  in  whole  numbers. 

EXAMPLES. 
How  much  in  whole  numbers  is 

(3.)  £|  (10.)  £.325. 


(4.)  tfton. 

(5.)  ^-mile. 

(6.)  |  yard. 

(7.)  f  acre. 

(8.)  f  hogshead. 

(9.)  ^  year. 


(11.)  .675cwt. 

(12.)  .75  rod. 

(13.)  .6  yard. 

(14.)  .25  sq.  mile. 

(15.)  .0025  tun. 

(16.)  .0785  day. 


CASE  III. 

Art*  123. — Reduction  of  Fractional  Compound  Numbers 
to  fractions  of  other  denominations. 

EXAMPLE  1. — Beduce  £^y  to  the  fraction  of  a  penny. 


FRACTIONAL  COMPOUND  NUMBERS.        163 

Process.  —  Reduction  descending  by  multiplication  of  common  frac- 
tions and  cancellation.     £^u=^  oX2OX12=fd.  ,  Ans. 

Ex.  2.  Reduce  f  oz.  to  the  fraction  of  a  cwt. 
Process.  —  "Reduction  ascending  by  division  of  common  fractions. 

OZ'  =     -  16' 


'  7Xl8*Xl 

Ex.  3.  Keduce  .00125  ton  to  pounds. 

.000125   ton. 

20 

Process.  —  Reduction  descending  by  multiplicatiou        .002500  cwt. 
of  decimals.  100 

-4ns.  .250000  Ib. 
Ex.  4.  Keduce  .5  rod  to  the  fraction  of  a  mile. 

Proeess.  —  Reduction  ascending  by  division  of    *  —  —  r°  S' 
decimals.  8).Q125  fur. 

Ans.  .0015625  mile. 

EXAMPU3S. 

5.  What  part  of  a  pound  is  f  of  an  oz.  Av.  ? 

6.  What  part  of  a  pound  is  y^j-  of  a  cwt.  ? 

7.  What  part  of  a  pound  is  .5  of  an  oz.  Troy  ? 

8.  What  part  of  a  pound  is  .0004  of  a  ton  ? 

9.  What  part  of  a  gallon  is  §  of  a  gill  ? 

10.  What  part  of  a  gallon  is  3-^  of  a  hhd.  ? 

11.  Wrhat  part  of  a  gallon  is  .25  of  a  pint  ? 

12.  What  part  of  a  gallon  is  .003  of  a  tun  ? 

13.  What  of  an  hour  is  ^  of  a  minute  ? 

14.  What  of  an  hour  is  -^  of  a  day  ? 

15.  What  of  an  hour  is  .8  of  a  second  ? 

16.  What  of  an  hour  is  .04  of  a  day  ? 

17.  What  part  of  a  rod  is  -^1  of  a  foot  ? 

18.  What  part  of  a  rod  is  ^  of  a  mile  ? 

19.  What  part  of  a  rod  is  .165  of  a  yard  ? 

20.  What  part  of  a  rod  is  .003  of  a  mile  ? 


164  FEACTTONAL  COMPOUND  NUMBERS. 

Art,  124. —  Addition  and  Subtraction  of  Fractional 
Compound  Numbers. 

EXAMPLE  1. — Add  £f  to  f  shilling. 

s.       d 
£|  =12        8 

Process.--  fs.=          10 

I3a      6d. 
Ex.  2.  From  £  cwt.  subtract  £  Ib. 

f  cwt.  =  3  qr.  8  Ib.  5^  oz. 
Process.—  f     Ib.  =  12 

Ans.  3  qr.  7  Ib.  9£    oz. 

Ex.  3.  Add  .6  acre  and  .06  rod. 

.6  acre. 
160 

Process     .6  ucre  =  96  rods,  to  which  add  .06  rods,      95,0      rods 
and  the  sum  will  be  96.06  rods.  .'06       « 

Ans,   96.06      " 

RULE. — Reduce  the  fractions  to  whole  numbers,  and  add 
or  subtract  them  as  other  compound  numbers. 

EXAMPLES. 

Ex.  4.  Add  f  of  a  day  to  f  of  an  hour. 

5.  From  £  of  a  mile  subtract  f  of  a  fur. 

6.  Add  .525  of  a  gal.  to  .9  of  a  qt. 

7.  Add  T%  Ib.  to  |  oz. 

8.  From  £  hhd.  take  £  of  a  barrel. 

9.  From  .625  of  an  hour  take  3.5  minutes. 


Art,  125.— Promiscuous  Examples  in  Fractional  Com- 
pound Numbers. 

EXEECISE   I. 

1.  How  much  in  whole  numbers  is  £%  ? 

2.  How  much  in  whole  numbers  is  .375  of  a  pound  (Troy)? 

3.  What  part  of  a  ton  is  5  cwt.  2  qr.  15  Ibs.  ? 


PROMISCUOUS  EXAMPLES.  165 

4.  What  part  of  a  hundred  weight  is  3  qr.  12  Ibs.  8  oz.  (in 

decimals). 

5.  What  part  of  a  drachm  is  T|¥  of  a  pound  (Apo.)? 

6.  What  part  of  a  yard  is  f  of  a  nail  ? 

7.  What  part  of  a  rod  is  .825  of  a  foot  ? 

8.  What  part  of  a  square  rod  is  .0025  of  an  acre  ? 

9.  What  is  the  sum  of  §  cord  and  §  foot  ? 

10.  How  much  more  is  .25  of  an  hour  than  5.5  minutes  ? 

EXEBCTSE  H. 

11.  How  much  in  whole  numbers  is  T6T  of  a  hogshead  ? 

12.  How  much  in  whole  numbers  is  .625  of  a  bushel  ? 

13.  What  part  of  a  year  is  125  days  12  hours  ? 

14.  What  part  of  a  degree  is  25'  36"  (in  decimals)  ? 

15.  What  part  of  a  farthing  is  y^  of  a  shilling  ? 

16.  What  part  of  a  pound  is  f  of  a  pwt.  ? 

17.  What  part  of  an  ounce  is  .84  of  a  scruple  ? 

18.  What  part  of  a  yard  is  .64  of  a  nail  ? 

19.  What  is  the  sum  of  .375  of  an  acre  and  75.5  rods  ? 

20.  How  much  more  is  f  of  a  rod  than  3^  yards  ? 

EXEKCISE  m. 

21.  How  much  in  whole  numbers  is  ^  of  a  mile  ? 

22.  How  much  in  whole  numbers  is  .865  of  a  cord  ? 

23.  What  part  of  a  barrel  is  21  gal.  ? 

24.  What  part  of  a  bushel  is  3  pks.  6  qts.  (decimal)  ? 

25.  What  part  of  a  second  is  y^j  of  a  minute  ? 

26.  What  part  of  a  degree  is  ^  of  a  circular  minute  ? 

27.  What  part  of  a  farthing  is  .0025  of  a  shilling  ? 

28.  What  part  of  an  ounce  is  .9  of  a  pwt.  ? 

29.  What  is  the  sum  of  %  ton  and  -g-  cwt.  ? 

30.  How  much  less  is  .26  cwt.  than  28.5  Ibs.  ? 

EXEBCISE  IV. 

31.  How  much  in  whole  numbers  is  |  of  a  pound  (A.)  ? 

32.  How  much  in  whole  numbers  is  2.5  of  a  yard  ? 

33.  What  part  of  a  yard  is  2  ft.  9  in..? 

34.  What  part  of  a  cord  is  21  cu.  ft.  576  in.  ? 

35.  What  part  of  a  quart  is  ^  of  a  gal.  ? 


166  FRACTIONAL  COMPOUND  NUMBEES. 

36.  What  part  of  a  peck  is  f  of  a  pint  ? 

37.  What  part  of  a  minute  is  .006  of  an  hour  ? 
33.  What  part  of  a  circle  is  .  72  of  a  degree  ? 

39.  Add  f  s.  6|  d.  2£  far. 

40.  How  much  more  is  .3  s.  than  2.25  d.? 

EXEECISE  v. 

41.  How  much  in  whole  numbers  is  £  of  a  pound  (Troy)  ? 

42.  How  much  in  whole  numbers  is  .0025  of  a  ton  ? 

43.  What  part  of  an  ounce  is  4  dr.  2  scr.  ? 

44.  What  part  of  a  yard  is  1  qr.  3  na.  (decimal)  ? 

45.  What  part  of  an  inch  is  T^  of  a  yard  ? 

46.  What  part  of  an  acre  is  |  of  a  rood  ? 

47.  What  part  of  a  cord  is  512  of  a  foot  ? 

48.  What  part  of  a  gill  is  .0056  of  a  gall.? 

49.  What  is  the  sum  of  .6  of  a  bushel  and  .8  of  a  peck  ? 

50.  How  much  more  is  ^  bushel  than  -g-  peck  ? 


Art.  126,— Promiscuous  Examples  in  Fractions,  Common 
and  Decimal,  and  Fractional  Compound  Numbers. 

EXEBCISE   I. 

1.  How  many  yards  are  there  in  three  pieces  of  cloth,  mea- 
suring as  follows:  30|,  37£,  38|  yards  ? 

2.  From  a  piece  of  cloth  which  contained  33£  yds.,  16f  yds. 
have  been  cut  off ;  how  many  are  left  ? 

3.  How  much  will  4^  tons  of  iron  cost  at  $18f  a  ton  ? 

4.  At  $18f  a  ton,  how  many  tons  of  iron  can  be  bought  for 
$60? 

5.  If  4|  tons  of  iron  cost  $75,  what  will  a  ton  cost  ? 

6.  How  many  are  3  and  7  tenths,  44  and  41  hundredths,  73 
and  9  thousandths,  12  and  305  thousandths  ? 

7.  Prom  5  hundredths  subtract  476  ten  thousandths. 

8.  At  $.1875  a  yard,  what  cost  12.25  yds.  of  muslin  ? 

9.  At  $.1875  a  yard,  how  many  yds.  of  muslin  can  be  bought 
for  $5.75  ? 


PROMISCUOUS   EXAMPLES.  167 

10.  If  19  yds.  of  muslin  cost  5.377,  what  is  the  price  per  yd.  ? 

11.  At  $18.875  a  1000,  what  will  12.500  feet  of  pine  boards 
cost? 

12.  At  $18.875  a  1000,  how  many  feet  of  pine  boards  can  be 
bought  for  $12.50  ? 

13.  If  1200  ft.  of  pine  boards  cost  $20,  what  is  the  price  per 
1000  ft.  ? 


14.  What  part  of  a  Ib.  Av.  is  ^  of  an  oz.  ? 

15.  What  part  of  a  nail  is  ^  of  a  yard  ? 

16.  What  part  of  a  mile  are  3  fur.  8  rods  ? 

17.  What  part  of  a  bushel  is  .5  of  a  peck  ? 

18.  What  part  of  a  pound  Troy  is  .  75  of  an  oz.  ? 

19.  What  part  of  a  pound  Troy  are  8  oz.  8  pwt.  (decimal)  ? 

20.  What  decimal  fraction  is  equal  to  %  ? 

21.  What  common  fraction  is  equal  to  .8  ? 

22.  What  whole  numbers  are  equal  to  T9^  of  a  day  ? 

23.  What  whole  numbers  are  equal  to  .5625  of  a  day  ? 

24.  How  much  are  %  of  a  week,  1%  days,  5>£  hours  ? 

25.  At  $.625  a  bushel,  what  cost  15  bus.  3  pks.  4  qts.  of  rye  ? 

26.  At  $.625  a  bushel,  how  much  rye  can  be  bought  for  $10? 

EXERCISE   II. 

27.  How  many  pounds  are  20}£,  21K,  22^,  23^,  24|  Ibs.? 

28.  From  a  cask  containing  64 1  gallons  of  molasses,    30^ 
gals,  have  been  used  ;  how  many  are  left  ? 

29.  At  $^  a  yard,  how  much  ribbon  can  be  bought  for 

$M? 

30.  At  $2^  a  yar(l,  n°w  niuch  will  3%  yards  of  ribbon  cost  ? 

31.  If  1%  yds.  of  ribbon  cost  $3^,  what  is  the  price  per  yd.  ? 

32.  How  many  are  44  and  19  thousandths,  8  and  71  hundred 
thousandths,   83  and  3327  ten  thousandths,  60  and  301  ten 
thousandths  ? 

33.  From  14  and  15  tenths  subtract  7  and  37  thousandths  ? 

34.  At  $12  a  ton,  how  much  coal  can  be  bought  for  $5.64  ? 

35.  At  $12  a  ton,  what  costs  3.047  tons  of  coal  ? 

36.  If  1.047  ton  of  coal  cost  $12.564,  what  is  the  price  per 
ton? 


168  FKACTIONAL  COMPOUND  NUMBERS. 

37.  At  $3.875  a  100,  how  many  bricks  can  be  bought  for 
$23.25? 

38.  At  $3.875  a  100,  how  much  wiU  3,750  bricks  cost  ? 

39.  If  4,575.  bricks  cost  $161.5625,  what  is  the  price  per  100  ? 


40.  What  part  of  a  pound  (Troy)  is  %  of  an  oz.  ? 

41.  What  part  of  an  oz.  is  yg^^  of  a  cwt.  ? 

42.  What  part  of  a  pound  (Troy)  are  9  oz.  12  pwt.  ? 

43.  What  part  of  a  yard  is  .8  of  a  nail  ? 

44.  What  part  of  a  bushel  are  3  pks.  2  qts.  (decimal)  ? 

45.  What  part  of  a  peck  is  .175  of  a  bushel  ? 

46.  What  decimal  fraction  is  equal  to  %  ? 

47.  What  common  fraction  is  equal  to  .16  ? 

48.  What  whole  numbers  are  equal  to  ff  of  a  mile  ? 

49.  What  whole  number  is  equal  to  .03515625  of  a  Ib.  Av.? 

50.  How  much  are  ^  hhd.  ,  \\  gal.  ,  and  \  qt.  ? 

51.  At  $4.80  a  cord,  how  much  wood  can  be  bought  for 
$70.80  ? 

52.  At  $4.80  a  cord,  what  cost  15  c.  96  ft.  of  wood  ? 

53.  If  13  cords  96  ft.  of  wood  cost  $61.875,  what  is  the  price 
per  cord  ? 

EXERCISE   HI. 

54.  How  many  are  28%,  f  of  18%,  32,  %  of  18%  ? 

55.  From  60  subtract  %  of  100£. 

56.  At  $}4  a  pound,  how  many  pounds  of  feathers  can  be 
bought  for  $8^  ? 

57.  At  $>£  a  Ib.,  how  much  will  15>£  Ibs.  of  feathers  cost  ? 

58.  If  141^  pounds  of  feathers  cost  $7>£,  what  is  the  price 
per  pound  ? 


59.  How  many  are  28  and  45-thousandths,  3  and  91-hun- 
dredths,  80  and  219  ten-thousandths,  17  and  7  tenths  ? 

60.  From  900  and   9-hundredths  subtract  99  and  9  thou- 
sandths. 

61.  At  $10.375  a  barrel,  how  many  barrels  of  flour  can  be 
bought  for  $72%  ? 

62.  At  $10.625  a  bbl.,  how  much  will  10  barrels  of  flour  cost  ? 


PROMISCUOUS  EXAMPLES.  169 

63    If  9  bbls.  of  flour  cost  $96.75,  what  is  the  price  per  bbl.  ? 

64.  At  $7.50  a  100,  how  many  cabbages  can  be  bought  for 
$5.625? 

65.  At  $7.50  a  100,  how  much  will  44  cabbages  cost  ? 

66.  If  56  cabbages  cost  $4.20,  what  is  the  price  per  100  ? 


67.  What  part  of  a  yard  is  f  of  a  nail  ? 

68.  What  part  of  an  inch  is  -fe  of  an  E.  English  ? 

69.  What  part  of  a  pound  Av.  are  9  oz.  2f  dr.  ? 

70.  What  part  of  a  yard  is  .5  of  a  quarter  ? 

71.  What  part  of  an  acre  is  .875  of  a  square  rod  ? 

72.  What  part  of  an  acre  are  1  rood,  14  rods  (decimal)  ? 

73.  What  whole  numbers  are  equal  to  f  of  a  yard  ? 

74.  What  whole  numbers  are  equal  to  .000175  of  an  acre  ? 

75.  From  f  of  an  ounce  subtract  %  of  a  pennyweight. 

76.  At  $.625  a  bushel,  how  much  corn  can  be  bought  for 
$9.00  ? 

77.  At  $.625  a  bushel,  what  cost  15  bu.  3  pks.  4  qts.  of  corn  ? 

78.  If  15  bu.  3  pks.  4  qts.  of  corn  cost  $9.525,  what  is  the 
price  per  bushel  ? 

EXERCISE  rv. 

79.  How  many  yards  in  six  pieces  of  cloth  measuring  as  fol- 
lows :  18%,  19*,  2%  20&,  22^,  24^  ? 

80.  From  a  piece  of  silk  which  contained  31%  yards,   8-f 
yds.  have  been  cut  off;  how  many  remain  ? 

81.  At  $|  a  pound,  how  much  tea  can  be  bought  for  $2.00  ? 

82.  At  $f  a  pound,  how  much  will  3|  Ibs.  of  tea  cost  ? 

83.  If  2^  Ibs.  of  tea  cost  $2,  what  is  the  price  per  pound  ? 


84.  Add  4  and  35  ten  thousandths,  10  and  35  hundred  thou- 
sandths, 6  and  35  millionths,  100  and  35  ten  millionths. 

85.  From  113  and  5  tenths  subtract  8  and  37  thousandths. 

86.  At  $45.625  a  barrel,  how  many  barrels  of  molasses  can 
be  bought  for  $136. 875? 

87.  At  $.375  a  gallon,  how  much  will  28.625  gal.  of  vinegar 
cost? 

8 


170  FRACTIONAL  COMPOUND  NUMBERS. 

83.  If  25.5  gals,  of  vinegar  cost  $8.925,  what  is  the  price  per 
gallon  ? 

89.  At  $21  a  1000,  how  much  will  150  shingles  cost  ? 

90.  If  200  shingles  cost  $3.50,  what  is  the  price  per  1000  ? 


91.  What  part  of  an  acre  is  %  of  a  square  rod  ? 

92.  What  part  of  a  minute  is  y^j  of  a  day  ? 

93.  What  part  of  an  A.  are  2.  B.  20  square  rods  ? 

94.  What  part  of  a  pint  is  .03125  of  a  gal.  ? 

95.  What  part  of  a  cwt.  are  1  qr.  8  Ibs.  10  oz.  (decimal)? 

96.  What  part  of  a  nail  is  .05  of  a  yard  ? 

97.  What  common  fraction  is  equal  to  .05  ? 

98.  What  decimal  fraction  is  equal  to  -^  ? 

99.  What  whole  numbers  are  equal  to  %  of  a  bushel  ? 

100.  What  whole  numbers  are  equal  to  to  .07  of  a  hhd.  ? 

101.  From  f  of  a  mile  subtract  -/y  of  a  fur. 

102.  At  27.25  an  acre,  how  much  land  can  be  bought  for 
$3500  ? 

103.  At  $105.  an  acre,  how  much  will  26  A.  2  E.  25  rods  of 
land  cost  ? 

104.  If  112  A.  3  E.  20  rods  of  land  cost  $11287.50,  what  is 
the  price  per  acre  ? 

EXEBCISE  V. 

105.  How  many  dollars  are  5£,  7^,  4%,  11%,  12%,  llf 
dollars  ? 

106.  From  f  of  16|£  subtract  4^. 

107.  At  4f  cts.  a  pound,  how  many  pounds  of  lead  can  be 
bought  for  $1.30f  ? 

108.  At  4|  cts.  a  pound,  how  much  will  28)^  Ibs.  of  lead 
cost? 

109.  If  23K  Ibs.  of  lead  cost  $1.45f ,  what  is  the  price  per  Ib.  ? 

110.  How  many  are  10  and  19  thousandths ;  106  and  3  hun- 
dredths ;  17  and  16  millionths ;  9  and  9  tenths  ;  71  and  63  ten 
thousandths  ? 

111.  From  51.004  subtract  31  and  8  hundredths. 


PROMISCUOUS  EXAMPLES.  171 

112.  At  $.347  a  pound,  how  much  will  9  Ibs.  of  tea  cost  ? 

113.  At  $.375  a  pound,  how  many  pounds  of  tea  can  be 
bought  for  $5.25  ? 

114.  If  11.23  pounds  of  tea  cost  $6.75,  what  is  the  price  per 
pound  ? 

115.  At  $5.625  a  100  feet,  how  many  feet  of  boards  can  be 
bought  for  $36.5625  ? 

116.  At  $5.625  a  100  feet,  how  much  will  1000  feet  of  boards 
cost? 

117.  What  part  of  a  pound  is  ||  of  a  pennyweight  ? 

118.  What  part  of  a  grain  is  -fa  of  a  dram  ? 

119.  What  part  of  a  bushel  are  1  peck,  5  qts. ,  1  pint  ? 

120.  What  part  of  a  pint  is  .025  of  a  gallon  ? 

121.  What  part  of  a  mile  are  110.4  rods  ? 

122.  What  part  of  a  mile  are  5  fur.  20  rods  (decimal)? 

123.  What  common  fraction  is  equal  to  .625  ? 

124.  What  decimal  fraction  is  equal  to  %  ? 

125.  What  whole  numbers  are  equal  to  f  of  a  mile  ? 

126.  From  %  of  a  hhd.  subtract  %  of  a  barrel. 

127.  At  $3.50  per  gal.,  how  much  will  37  gals.,  2  qts.,  1  pt. 
of  wine  cost  ? 

128.  If  25  gals.,  1  qt.,  1  pt.  of  wine  cost  $78.6625,  what  is 
the  price  per  gallon  ? 

\-  EXERCISE  YL 

129.  How  many  gals,  are  8%,  11%,  9|,  6^,  1Q&,  6|  gals.? 

130.  A  man  having  137%  acres,  sold  25f ,  how  many  acres 
had  he  left  ? 

131.  At  $26^  an  acre,  how  many  acres  of  land  can  be 
bought  for  $1000  ? 

132.  At  $26^  an  acre,  what  will  33>£  acres  of  land  cost  ? 

133.  If  33>£  acres  of  land  cost  $1666%,  what  is  the  price  per 
acre? 


134.  What  is  the  sum  of  9  and  9  tenths ;  10  and  12  thous- 
andths ;  100  and  1  hundredths  ;  1000  and  1  thousandths ;  10000 
and  1  ten  thousandths  ? 


172  FRACTIONAL   COMPOUND   NUMBERS. 

135.  From  300.  subtract  3  Imndredths. 

136.  At  SI.  75  a  bushel,  how  many  bushels  of  peaches  can  be 
bought  for  $25  ? 

137.  At  $1.75  a  bushel,  how  much  wiU  20^   bushels  of 
peaches  cost  ? 

138.  If  20}£  bushels  of  peaches  cost  $36.90,  what  is  the  price 
per  bushel  ? 

139.  At  $7.20  a  1000,  what  cost  19.625  bricks  ? 

140.  At  $7.20  a  1000,  how  many  bricks  can  be  bought  for 
$148.50  ? 

141.  If  20000  bricks  cost  $140,  what  is  the  price  per  $1000  ? 

142.  What  part  of  a  barrel  is  -ffi,  of  a  gill  ? 

143.  What  part  of  a  nail  is  ^  of  a  yard  ? 

144.  What  part  of  a  cubic  foot  is  .0005  of  a  cord  ? 

145.  What  part  of  an  acre  are  3  E.  14  sq.  rds.  16  sq.  yds.  4 
sq.  ft.  and  72  sq.  in.  ? 

146.  What  part  of  a  cwt.  are  3  Ibs.  11  oz.  3.2  dr.  (decimal)? 

147.  What  common  fraction  is  equal  to  .35  ? 

148.  What  decimal  fraction  is  equal  to  f  ? 

149.  What  whole  numbers  are  equal  to  .375  of  an  ounce  ? 

150.  What  whole  number  is  equal  to  f  of  a  month  ? 

151.  Add  %  of  a  ton  and  £  of  a  cwt. 

152.  At  $4  a  yd.,  how  much  cost  4  yds.  3  qrs.  1  na.    of 
cloth  ? 

153.  At  $5.25  a  yard,  how  much  cloth  can  be  bought  for 
$34.125? 

EXEECISE  VH. 

154.  How  many  miles  are  7>£,  3}£,  6)£,  4£,  8T^,  12^  miles  ? 

155.  Sold  a  carriage  for  $175}^,  and  gained  $14%  ;  what  did 
it  cost  ? 

156.  At  $4)^  a  bushel,  how  much  will  7f  bushels  of  clover 
seed  cost  ? 

157.  At  $4}<j  a  bushel,  how  many  bushels  of  clover  seed  can 
be  bought  for  $39%  ? 

158.  If  6f  bushels  of  clover  seed  cost  $30%,  what  is  the 
price  per  bushel  ? 


PROMISCUOUS   EXAMPLES.  173 

159.  What  is  the  difference  between  3  hundredths  and  3 
thousandths  ? 

160.  At  $.1575  a  pound,  how  much  sugar  can  be  bought  for 
$15.04125? 

161.  At  $.15  a  pound,  how  much  will  .375  of  a  pound  of 
sugar  cost  ? 

162.  If  21.5  pounds  of  sugar  cost  $3.225,  what  is  the  price 
per  pound  ? 

163.  At  $5.50  a  100  Ibs.,  how  much  flour  can  be  bought  for 
$22? 

164.  At  $5.75  a  100  Ibs.,  how  much  will  637.5  pounds  of 
flour  cost  ? 

165.  If  531.75  pounds  of  flour  cost  $35.159,  what  is  the  price 
per  100  Ibs.? 

166.  What  part  of  a  rod  is  ^f  of  an  inch  ? 

167.  What  part  of  a  dram  is  7^  of  a  pound  ? 

168.  What  part  of  a  ban-el  are  8  gals.  1  qt.  3f  gills  ? 

169.  What  part  of  a  grain  is  .00075  of  an  oz.,  Aps.  ? 

170.  What  part  of  a  pound,  Troy,  is  .432  of  a  grain  ? 

171.  What  part  of  a  degree  are  41'  51"  (decimal)? 

172.  What  common  fraction  ==  .625  ? 

173.  What  decimal  fraction  =  ^  ? 

174.  What  whole  numbers  =  .037625  of  ton  ? 

175.  What  whole  numbers  =  %  of  a  tun  ? 

176.  From  %  ton  subtract  f  cwfc. 

177.  At  $.08  a  pound,  how  much  will  12  Ibs.  5  oz.  01  soap 
cost? 

178.  At  $075  a  pound,  how  much  soap  can  be  bought  for 
$3.00? 

179.  If  75  pounds  of  soap  cost  $6.,  what  is  the  price  per  lb.? 

EXERCISE  vm. 

180.  How  many  acres  are  38|,  45  jj,  50$.  35^-  acres  ? 

181.  Bought  a  piece  of  cloth  containing  27£  yards.     I  have 
sold  £  of  it,  how  much  is  left  ? 

182.  At  2£  cents  a  foot  how  many  feet  of  lumber  can  be 
bought  for  $1  ? 


PKOMISCUOUS  EXAMPLES. 

183.  At  2f  cents  a  foot,  how  much  will  19^  feet  of  lumber 
cost  ? 

184.  If  15£  feet  of  lumber  cost  35f  cents,  what  is  the  price 
per  foot  ? 

185.  What  is  the  difference  between  7  thousandths  and  7 
millionths  ? 

186.  At  $19.875  a  barrel,  how  many  barrels  of  pork  can  be 
bought  for  $139. 125? 

187.  At  $19.875  a  barrel,  how  much  will  10  barrels  of  pork 
cost  ? 

188.  If  8  barrels  of  pork  cost  $159,  what  is  the  price  per 
barrel  ? 

189.  At  $11.50  a  100  pounds,  how  much  will  210  Ibs.  of 
sugar  cost  ? 

190.  If  256  pounds  of  sugar  cost  $29.44,  what  is  the  price 
per  100  Ibs. 

191.  What  part  of  a  yard  is  -^  of  a  nail  ? 

192.  What  part  of  a  grain  is  -^^  of  a  Ib.  Troy  ? 

193.  What  part  of  a  cord  are  26  cubit  feet  1152  inches  ? 

194.  What  part  of  a  quart    is  .0005  hhd.  ? 

195.  What  part  of  a  mile  are  4  furlongs  30  rods  2  yards  2 
feet,  3  inch,  decimal  ? 

196.  What  common  fraction— .075  ? 

197.  What  decimal  fraction^  £  ? 

198.  What  whole  numbers=.15  of  a  bushel  ? 

199.  What  whole  numbers=.f  of  an  hour  ? 

200.  From  11  of  £1,  subtract  7s.  8d. 

201.  At  $13  a  ton,  how  much  hay  can  be  bought  $75  ? 

202.  At  $13  a  ton,  how  much  will  18  cwt.  1  qr.  21  Ibs.  of 
hay  cost  ? 

203.  If  18  cwt.  1  qr.  21  Ibs.  cost  $11.50,  what  is  the  price 
per  ton  ? 

EXERCISE  IX. 

204.  How  many  yards  in  three  pieces  of  calico,  measuring 
24,  21J,  20^  yds.  ? 

205.  At  $f-  a  bushel,  how  many  bushels  of  potatoes  can  be 
bought  for  $5*  ? 


PROMISCUOUS   EXAMPLES.  175 

206.  At  $1£  each,  how  much  will  9  sheep  cost  ? 

207.  A  man  having  9|  cords  of  wood,  has  sold  3^  cords  ;  how 
much  has  he  left  ? 

208.  At  $1|-  a  day,  how  much  will  a  man  earn  in  26^  days  ? 

209.  From  1  hundred   and  1,  subtract  1  hundred    and  1 
hundredths  ? 

210.  At  $.1875  a  pound,  what  will  10.4  Ibs.  of  beef  cost  ? 

211.  At  $2.75  a  hundred  feet,  how  many  feet  of  timber  can 
be  bought  for  $736.945  ? 

212.  At  $2.25  a  hundred,  what  will  175.28  feet  of  timber 
cost? 

213.  At  $2  a  bushel,  how  many  peaches  can  be  bought  for 
37£  cents  ? 

214.  At  $2  a  bushel,  what  will  3  pecks,  5  quarts,  of  peaches 
cost? 

215.  At  $5  a  square  rod,  what  will  a  village  lot  cost,  contain- 
ing ^  acre  ? 

216.  If  a  man  can  earn  $1  in  £  of  a  week,  how  much  does  he 
earn  in  a  day  ? 

217.  If  f  of  an  ounce  of  nutmegs  cost  10  cents,  what  is  the 
price  per  pound  ? 

218.  Three  village  lots  contain  respectively  1^  acres,   3£ 
roods,  30^  rods  ;  how  much  do  they  all  contain  ? 

219.  At  $8.95  a  cwt.,  how  much  sugar  can  be  bought  for 
$43.855  ? 

220.  At  $8.75  a  cwt.,  how  much  will  2  cwt.  1  quarter  15  Ibs. 
of  sugar  cost  ? 

221.  If  3  cwt.  2  quarters  10  Ibs.  of  sugar  cost  $27,  what  is 
the  price  per  cwt  ? 

EXERCISE  X. 

222.  At  £  of  a  cent  each,  how  many  apples  can  be  bought 
for  56  cents  ? 

223.  If  65  apples  cost  56£  cents,  how  much  does  1  apple 
cost? 

224.  At  |  cents  each,  how  much  cost  60  apples  ? 

225.  How  many  yards  are  left  in  a  piece  of  cloth  which  con- 


176  PKOMISCUOUS   EXAMPLES. 

tained  37  yards,  after  cutting  off  at  different  times  5J,  3f  ,  8§, 
7£  yards? 

226.  At  $6f  a  ton,  what  cost  8|  tons  of  hay  ? 

227.  If  9|  tons  of  hay  cost  $63f  ,  what  is  the  price  per  ton  ? 

228.  At  $6f  a  ton,  how  many  tons  of  hay  can  be  bought  for 


229  From  one  thousand  and  seven  hundredths,  subtract  five 
hundred  and  five  hundredths. 

230.  What  cost  7.625  cords  of  wood  at  $6.50  a  cord  ? 

231.  At  $8.50  a  cord,  how  many  cords  of  wood  can  be  bought 
for  $61.  75? 

232.  If  7.625  cords  of  wood  cost  $61,  what  is  the  price  per 
cord  ? 

233.  At  $9.50  a  ton  what  will  12  cwt.  40  Ibs.  of  coal  cost  ? 

234.  At  $12  a  barrel,  how  much  oil  can  be  bought  for  $9.  75  ? 

235.  If  3)^  pints  of  oil  cost  50  cts.  what  is  the  price  per  gal.  ? 

236.  If  .5  hhd.  of  molasses  cost  $15.75  what  ia  the  price  of 
a  quart  ? 

EXERCISE   XI. 


237.  A  lady  bought  %  of  a  piece  of  muslin  containing 
yds.  ;  %  of  another  containing  33)^  yds.  ;  ^  of  another  con- 
taining 44j^  yds.  ;  and  a  whole  piece  containing  31%  yds.  ;  how 
many  yards  did  she  buy  ? 

238.  At  5%  cts.  a  pound,  how  much  will  a  calf  weighing 
137^  Ibs.  cost  ? 

239.  At  7%  cts.  a  pound  how  much  veal  can  be  bought  for 
87^  cts.? 

240.  If  a  calf  weighing  118%  Ibs.  cost  $12.41%  what  is  the 
price  per  pound  ? 

241.  Bought  a  horse  for  $112.5625  and  sold  him  for  $125, 
how  much  was  gained  ? 

242.  At  $2.6875  a  yard,  how  much  cassimere  can  be  bought 
for  $43.671875  ? 

243.  At  $2.6875  a  yard,  what  cost  16.25  yds.  of  cassimere  ? 

244.  If  16.25  yds.  of  cassimere  cost  $40,  what  is  the  price 
per  yard  V 


PROMISCUOUS   EXAMPLES.  177 

245.  At  $4.75  a  100,  how  many  shingles  can  be  bought  for 
$36. 29? 

246.  At  $25  a  1000  what  cost  9875  shingles  ? 

247.  If  1200  shingles  cost  $24.75  what  cost  100  ? 

248.  At  $.125  a  yd.,  what  cost  11  yd.  2  qr.  3  na.  of  muslin  ? 

249.  If  10  yd.  1  qr.  of  muslin  cost  $1.64,  what  is  the  price 
per  yard  ? 

250.  At  $.125  a  yard,  how  many  yards  of  muslin  can  be 
bought  for  $5. 25? 

EXERCISE  XIL        x 

251.  How  much  carpeting  %  yd.  wide  will  cover  a  room  22)^ 
ft.  long,  18  ft.  wide  ? 

252.  How  many  pieces  of  paper  \%  ft.  wide  and  8)^  "yds. 
long  must  be  bought  to  cover  the  sides  of  a  room  18%  ft.  long, 
16)^  ft.  wide,  and  11^  ft.  high  ;  there  being  one  door  6%  ft. 
by  2%,  and  two  windows  5>£  by  2^  ft.  ? 

253.  How  much  will  it  cost  to  pave  a  street  ^  mile  long  and 
2)£  rods  wide,  at  $12  a  square  rod  ? 

254.  If  a  piece  of  land  containing  5.5  acres  be  divided  into 
building  lots  4  rods  long  and  2.2  rods  wide,  what  would  they 
all  be  worth  at  $200  each  ? 

255.  How  many  shingles  will  cover  the  roof  of  a  house  32  ft. 
long  the  rafters  on  each  side  being  16%  ft.  long,  allowing  one 

"  shingle  for  every  24^  sq.  in.  ? 

256.  How  much  will  it  cost  to  dig  a  ditch  around  a  garden 
6.5  rods  square,  the  ditch  to  be  3.25  feet  deep  and  2.5  feet  wide, 
atl  ct,  a  cubic  foot  ? 


DUODECIMALS. 

Arti  127. — Duodecimals  are  a  species  of  fractional  com- 
pound numbers  sometimes  used  in  measuring  lumber, 
&c.  They  arise  from  successive  divisions  of  1  foot  by  12. 
(Latin,  duodecim). 


178  DUODECIMALS. 

TABLE. 

12""  (fourths)  make 1"  (third)  =  r^  ft. 

12'"  (thirds)  1"  (second)  =    T^  ft. 

12"    (seconds)  1'    (prime,  or  inch)  =     ^-  ft. 

12'    (inches)  1     (foot) 

The  marks  used  to  distinguish  the  different  denomina- 
tions are  called  Indices* 

Duodecimals  may  be  added,  subtracted,  multiplied  and 
divided,  like  other  compound  numbers. 

In  multiplication  of  duodecimals  by  duodecimals — 

Feet  multiplied  by  feet  give  square  feet. 

Square  feet  multipled  by  feet  give  cubic  feet. 

Feet  multiplied  by  inches  (as  1XA)  give  sq.  inches,  ^  sq.  ft. 

Square  feet  multiplied  by  inches  (as  iXiV)  giye  inches,  ^ 
cubic  foot. 

Inches  multiplied  by  inches  (as  -^XA)  give  square  inches, 
or  seconds,  T|?  square  foot. 

Square  inches  multiplied  by  inches  (as  T^XT^)  give  cubic 
sq.  in.  or  seconds,  T|^  cu.  ft. 

Inches  multiplied  by  seconds  (as  iVXyl?)  giye  thirds  d?1^) 
sq.  foot. 

Square  inches  multipled  by  seconds  give  thirds  (77^)  cu.  ft. 

Seconds  multiplied  by  seconds  give  fourths,  &c. 

EXAMPLE  1. — What  are  the  contents  of  a  board  10  feet  6 

inches  long,  and  2  feet  3  inches  wide  ? 

ft.  in. 

10  6 

Process -6X3'=^XA=iL4V=18''=r  6".      Write     J_ 

6"  and  carry  1'.     Next  10  feet  X  3'=10X-&=ff  and       2  76 

1'  makes  ft=Zfa=&  feet  7  inches.    Then  6'X5  ft. ,  &c.      52  6 

55  ft.  1'  6" 

RULE. — Multiply  each  term  in  the  multiplicand  by  each 
term  in  the  multiplier,  giving  each  product  an  index  equal  to 
the  indices  of  both  its  factors  ;  then  after  reducing  and  carry- 
ing, as  in  compound  numbers,  add  the  like  terms  of  the  pro- 
ducts. 


DUODECIMALS.  179 

EXAMPLES. 

(2.)  Multiply  8  feet  four  inches  by  3  feet  9  inches. 


(3.)  9  ft.  6  in.  by  2  ft.  8  in. 

(4.)  12  ft.  10  in.  by  4  ft.  3  in. 

(5.)  10  ft.  8  in.  by  5  ft.  2  in. 

(6.)  15ft.  5  in.  by  3  ft.  4  in. 


(7.)  13ft.  Tin.  by  6  ft.  5  in. 

(8.)  14ft.  8  in.  by  7  ft.  lin. 

(9.)  lift.  9  in.  by  9  ft.  3  in. 

(10.)  16ft.  4  in.  by  3  ft.  3  in. 


11.  How  many  sqiiare  feet  are  there  in  a  board  14  feet  9 
inches  long,  and  2  feet  wide  ? 

12.  How  many  square  feet  in  a  board  16  feet  8  inches  long, 
and  1  foot  10  inches  wide  ? 

13.  How  many  square  feet  in  a  door  6  ft.  6  inches  long,  and 
3  feet  4  inches  wide  ? 

14.  How  many  square  feet  in  a  floor  18  feet  10  inches  long 
and  15  feet  wide  ? 

15.  How  many  square  feet  in  a  piece  of  molding  20  feet  long 
and  3  inches  wide  ? 

16.  How  many  square  feet  in  20  boards,  each  12  feet  long 
and  9  inches  wide  ? 

17.  How  many  cubic  feet  in  a  stick  of  timber  10  feet  long, 
1  foot  3  inches  wide,  and  4  inches  thick  ? 

18.  How  many  cubic  feet  in  a  load  of  wood  8  feet  long,  4 
feet  6  inches  high,  and  3  feet  10  inches  wide  ? 

19.  How  many  cubic  feet  in  a  block  of  marble  6  feet  8  inches 
long,  2  feet  6  inches  wide,  and  2  feet  thick  ? 

20.  How  many  cubic  feet  in  a  wall  21  feet  6  inches  long,  6 
feet  3  inches  high,  and  2  feet  thick  ? 

21.  What  will  it  cost  to  plaster  a  room  24  feet  6  inches  long, 
15  feet  5  inches  wide,  and  8  feet  4  inches  high,  at  36  cents  a 
square  yard  ? 

22.  How  many  bricks  8  inches  long,  4  inches  wide,  and  2 
inches  thick,  will  it  take  to  build  a  wall  72  feet  long,  4  feet  6 
inches  high,  and  1  foot  thick,  supposing  the  bricks  not  to  be 
separated  by  mortar  ? 


180  ANALYSIS. 

ANALYSIS. 

Art.  128.  —  Analysis  in  Arithmetic  is  a  method  of  solv- 
ing questions  without  formal  rules.  Rules  are  derived 
from  analysis.  The  process  consists  generally  in  reason- 
ing from  a  given  number  to  1  of  the  same  kind,  and 
from  1  to  the  required  number. 

EXAMPLE  1.—  If  7  pounds  of  sugar  cost  $1.12,  how  much  will 
42  pounds  cost  ? 

Process.—  If  7  Ibs.  of  sugar  cost  $1.12,  1  Ib.  will  cost  |  of  $1.12,  or 
16  cts.,  and  42  Ibs.  will  cost  42  times  16  cts.,  or  $6.72,  Ans. 

Or  since  42  Ibs.  is  6  times  7  Ibs.,  42  Ibs.  will  cost  6  times  $1.12 
(the  price  of  7  Ibs.),  and  $1.  12X6=$6.  72,  Ans. 

Ex.  2.  If  |  of  a'  yard  of  cloth  cost  $4£,  how  much  will  I  of 
a  yard  cost  ? 

Process.—  If  £  yard  cost  ($4£)  $|,  £  yard  will  cost  $f,  and  f,  or  1 
yard,  will  cost  $6.  Then  £  yard  will  cost  $|,  and  I  will  cost  ±£,  or 
$  4|,  Ans. 

In  examples  like  the  last  it  is  better  to  express  each  multiplication 
by  writing  the  multiplier  as  a  factor  in  the  numerator  of  a  fraction, 
and  each  division  by  writing  the  divisor  as  a  factor  in  the  denomina- 
tor, then  cancel,  &c.,  thus  : 
2 


Ex.  3.  Barter.  —  How  many  loads  of  wood  at  $4£  will  pay  for 
3  barrels  of  flour  at  $8f  ? 

Process.—  If  1  bbl.  of  flour  costs  $8f,  3  bbls.  will  cost$2o|-,  and  if 
$4£  will  pay  for  1  load  of  wood,  $26£  will  pay  for  as  many  loads  as 
$4|r  is  contained  times  in  $26^,  which  is  6.  Ans.,  6  loads. 

Ex.  4.  Aliquot  parts  or  Practice.  —  What  cost  5  cwt.  65  Ibs., 
at  £2  5s.  6d.  per  cwt.  ? 

Process.—  50  lbs.=i  cwt.  £2    5s.    6d. 

5 


1176     price  of  5  cwt. 

10  Ibs.  =±  of  50  Ibs.          129         "        50  pounds. 
5  Ibs.  =i  of  10  Ibs.  4     6f       »         10       " 

2     3-,3q      «          5       " 

Ans..  12  17     Oft 


ANALYSIS.  181 

Ex.  5.  A  general  lost  %  of  his  army  in  battle,  %  were  taken 
prisoners,  £  deserted,  and  he  had  2600  men  left ;  how  many 
had  he  at  first  ? 

Process.— iXiXi=$fc  »nd  tne  remainder  ££=2600.  &=200,  18= 
12000,  the  army  at  first. 

Analysis  may  also  be  applied  to  questions  under  rules  to  be  here- 
after given,  such  as  Proportion,  Partnership,  Reduction  of  Currencies, 
Alligation,  &c. 

EXAMPLES. 

6.  If  20  barrels  of  apples  cost  $50,  what  will  35  bbls.  cost  ? 

7.  If  33  tons  of  coal  cost  $198,  how  much  will  11  tons 
cost? 

8.  If  15  pounds  of  butter  cost  $4.50,  how  much  will  70  Ibs. 
cost? 

9.  If  12  pairs  of  shoes  cost  $30,  how  much  will  48  prs.  cost  ? 

10.  How  much  will  65  sheep  cost  if  5  sheep  cost  $17.50  ? 

11.  How  much  are  50  cows  worth,  if  10  cows'  are  worth 
$450? 

12.  If  %  of  an  acre  of  land  cost  $66%,  how  much  will  6% 
acres  cost  ? 

13.  How  much  will  f  of  a  ton  of  hay  cost,  if  f  of  a  ton  costs 
$12? 

14.  How  much  will  ^-  of  a  cord  of  wood  cost,  if  ^-  costs 
$1.12^  ? 

15.  If  f  of  a  pound  of  tea  costs  $^,  how  much  will  f  of  a 
pound  cost  ? 

16.  If  f  of  a  yard  of  cloth  costs  $4.80,  how  much  will  £  of  a 
yard  cost  ? 

17.  If  |  of  a  cord  of  wood  cost  $1.10,  how  much  will  f  of  a 
cord  cost  ? 

18.  How  much  will  ^  of  a  ton  of  plaster  cost  if  f  of  a  ton 
'    9 


19.  How  much  will  f£  of  an  acre  of  land  be  worth  if  ^  of 
an  acre  is  worth  $30  ? 

20.  How  much  will  if  of  a  chain  30  feet  long  cost  if  §  of  a 
like  chain  36  feet  long  is  worth  $24  ? 


182  ANALYSIS. 

21.  How  many  eggs  at  20  cents  a  dozen  must  be  given  for  9 
pounds  of  butter  at  30  cents  a  pound  ? 

22.  How  many  pounds  of  lard,  at  15  cents  a  pound,  will  pay 
for  14  pounds  of  sugar,  at  12)£  cents  a  pound  ? 

23.  How  many  bushels  of  oats,  at  37^  cents  a  bushel,  will 
pay  for  5  yards  of  cloth,  at  $4.50  ? 

24.  How  many  yards  of  calico,  at  18%  cents  a  yard,  can  be 
bought  for  10  pounds  of  butter,  at  31)^  cents  ? 

25.  At  $87.50  an  acre,  what  cost  8  acres  110  rods  ? 

26.  At  $6.75  a  ton,  what  cost  7  tons  12  cwt.  60  Ibs.  of  hay  ? 

27.  At  $5.37^  a  yard,  what  cost  3  yards  3  quarters  3  nails 
of  cloth  ? 

28.  At  £2  11s.  6d.  a  bushel,  what  cost  5  bushels,  1  peck,  4 
quarts  of  timothy  seed  ? 

29.  At  £10  12s.  6d.  an  acre,  what  cost  9  acres  60  rods  ? 

30.  At  £2  8s.  6^d.   a  cwt.,  what  cost  7  cwt.  30  pounds  of 
flour? 

31.  A  regiment  of  soldiers  was  diminished  ^  by  sickness,  £ 
captured  by  the  enemy  ;  ^  killed  and  missing,  and  then  con- 
sisted of  250  men  ;  how  many  did  it  number  at  first  ? 

32.  A  young  man  spent  -|  of  his  property  in  3  years,  ^  of  it 
the  next  2  years,  and  then  he  had  $600  left ;  how  much  had  he 
at  first  ? 

33.  Paid  $2,100  for  ^j  of  a  vessel,  what  was  the  whole  vessel 
worth? 


34.  If    18  Ibs.   of    cheese  cost  $2.70,  what  will  a  cheese 
weighing  72  Ibs.  cost  ? 

35.  If  f  of  hogshead  of  sugar  cost  $34,  what  will  f  of  a 
hogshead  cost  ? 

36.  If  £  of  a  gallon  of  alcohol  cost  $3f ,  what  will  T\  of  a 
gallon  cost  ? 

37.  How  many  bushels  of  corn  at  90  cents  a  bushel,  will  pay 
for  63  Ibs.  of  beef  at  14  cents  a  pound  ? 

38.  At  $5.38  a  cwt.,  how  much  will  60  Ibs.  of  flour  cost  ? 

39.  If  a  barrel  of  ale  cost  £7  14s.  4d.,  what  will  24  gallons 
1  quart  cost  ? 


ANALYSIS.  183 

40.  There  is  a  town  in  which  £  the  men  are  farmers,  £  me- 
chanics, |  laborers,  and  the  rest  26  without  employment ;  how 
many  men  are  there  in  the  town  ? 


41.  How  many  pounds  of  pork  at  12^  cents  a  pound,  will 
pay  for  7  days  labor,  at  $1.121  a  day  ? 

42.  If  30  pounds  of  coffee  cost  $7.50,  what  will  9  Ibs  cost. 

43.  If  %  of  a  cord  of  wood  is  worth  $3£,  how  much  is  j~|  of 
a  cord  worth  ? 

44.  How  much  will  £  of  a  pound  of  soap  cost,  if  f  of  a 
pound  cost  12  cents  ? 

45.  At  $94  an  acre,  what  will  50  square  rods  cost  ? 

46.  How  much  will  29  gals,  of  vinegar  cost  at  $10}^  a  barrel  ? 

47.  A  man  left  his  elder  son  %  of  his  property,  the  younger 
YD  and  the  elder  son  had  $1000  more  than  the  younger  ;  what 
was  the  whole  property? 


48.  If  %  of  a  ship  is  worth  $56,000,  what  is  ^  of  it  worth  ? 

49.  What  is  \  of  a  bushel  of  clover  seed  worth,  if  f  of  a 
bushel  is  worth  $2f  ? 

50.  If  a  man  earn  $42  in  12  days,  how  much  will  he  earn  in 
50  days  ? 

51.  How  many  days'  labor  at  $.87>£  a  day  will  pay  for  7 
bushels  of  buckwheat  at  75  cents  a  bushel  ? 

52.  At  $21. 31 1£  a  cwt.,  how  much  will  86  Ibs.  of  honey  cost  ? 

53.  If  a  year's  labor  is  worth  £100,  how  much  will  it  be  for 
7  months  and  20  days  ? 

54.  A  benevolent  lady  gave  %  of  her  income  to  the  Bible 
Society,  and  f  of  it  to  the  poor,  reserving  only  $600  for  herself, 
what  was  her  income  '? 

55.  If  |  of  a  ton  of  hay  cost  f  of  $9%,  how  much  will  ^  of 
a  ton  cost  ? 

56.  How  many  barrels  of  apples  at  $2)^  a  barrel,  will  pay 
for  4  bbls.  of  flour  at  $9^? 

57.  What  will  ^  of  a  yard  of  velvet  cost,  if  f  of  a  yaid  cost 
$1.08  ? 


184  PERCENTAGE. 

58.  What  will  15  yards  of  calico  cost,  if  9  yds.  cost  $1.62  ? 

59.  At  $4.50  a  yard,  what  will  3  yds.  1  qr.  3  na.  cost  ? 

60.  A  lady  gave  )^  of  her  property  to  her  son,  yz  to  her 
daughter,  and  the  rest,  amounting  to  $1,200,  to  benevolent 
objects,  what  was  the  amount  of  her  property  ? 


PERCENTAGE, 

Art.  129.— Percentage  is  calculating  numbers  by  hun- 
dredths,  or  parts  of  a  hundred. 

Per  cent,  (derived  from  the  Latin  words  per  centum,, 
meaning  by  the  hundred)  is  used  in  expressing  hun- 
dredths,  or  parts  of  a  hundred  ;  thus,  5  per  cent,  is  5 
hundredths,  or  five  for  every  hundred  (dollars,  pounds, 
&c. ) ;  6  per  cent,  is  6  hundredths. 

The  sign  %  is  often  used  for  per  cent. 

Art.  130. — In  Percentage  three  things  are  chiefly  con- 
sidered. 

The  Principal,  the  number  on  which  percentage  is 
calculated. 

The  Rate  per  cent.,  the  number  of  hundredths. 

The  Percentage,  the  number  which  the  principal  pro- 
duces at  a  given  rate. 

Any  two  of  these  being  known,  the  other  maybe  found. 

The  term  Principal  thus  used  includes,  but  is  not  limited  to  money 
at  interest. 

The  rate  per  cent,  is  expressed  by  a  fraction,  usually  a 
decimal  fraction,  thus: 

1  per  cent,  is  written 01  =  y^ 

5  per  cent,  is  written .05  =  y^  or  ^ 

10  per  cent,  is  written 10  =  ^ 

25  per  cent,  is  written 25  =   4 


PERCENTAGE.  185 

100  per  cent,  is  -written 1.00  =  the  whole. 

y2  per  cent,  is  written 005  or  .00>£. 

%  per  cent,  is  written 0025  or  .00^. 

Some  rates  per  cent,  cannot  be  exactly  expressed  by  decimals  ;  as 
^  per  cent,  must  be  written  .00^  ;  33^  per  cent.  .33^. 

Write  the  following  rates  per  cent. 

3  per  cent.;  6,  4,  12,  7,  8,  15,  9,  20,  #,  %,  %,  J,  %,  2^, 
8X,  18%,  3f,  %,  37^,  i  33^,  %,  ™%,  6#,  12^,  10|,  75, 
110,  125,  137%. 

CASE  I. 
Art  t  131. — To  find  the  percentage. 

MENTAL  EXERCISES. 

How  much  is  3  per  cent,  of  $5  ? 

Process.— Since  3  %  is  .03,  3  %  of  $5.  is  .03  times  $5,  which  is  .15 
cents.  Therefore  3  %  of  $5.  is  15  cents. 

How  much  is  4  %  of  $1  ?  $10  ?  $12  ?  $8  ?  50  cents  ? 
How  much  is  5  %  of  4  pounds  ?  20  Ibs.  ?  50  Ibs.  ?  100  Ibs.  91 
If  a  miller  take  4%  toll  for  grinding. wheat,  how  much  will 
it  be  for  grinding  100  bushels  ?  200  ?  50  ?  25  ? 

EXAMPLES  FOB  THE   SLATE. 

EXAMPLE  1.  What  is  6^  %  of  568  pounds  ? 

568 
.0625 

Process.— 6£  %  =  .0625,  therefore  6£  %  of  568  — ooJn 

Ibs.  =  568  X-0625  =  35. 5000  =  35£  Ibs.  1:136 

3408 
^ns735T5000  Ibs. 

Or  <*K=i$  =  n  «*  i  of  568=351. 

RULE. — Multiply  the  principal  by  the  rate  per  cent. 
Percentage  =  Principal  X  Rate. 

2.  What  is  3  %  of  $8750  ?   6^?  7%? 

3.  What  is  5  %  of  364  gallons  ?  7  %"?  12  %  ? 

4.  What  is  6%  of  576  pounds?  8%  ?   16%? 

5.  What  is  7  %  of  368  bushels  ?  9  %  ?  4)^  %  ? 

6.  What  is  8K  %  of  261  gallons  ?  f  %  ?  16%  %  ? 


186  PERCENTAGE. 

7.  A  farmer  has  320  bushels  of  wheat,  25  %  more  of  oats, 
and  12)^  %  less  of  corn  ;  how  many  bushels  of  oats  has  he  ? 
Of  corn  ? 

8.  A  regiment  consisted  of  840  soldiers,  of  whom  16%  were 
killed  and  missing  ;  how  many  were  left  ? 

9.  A  merchant  having  $6400  capital,  gained  18%  % ;  how 
much  did  he  gain  ? 

10.  Another  merchant,  having  $7200  capital,  lost  12}£  %  of 
it ;  how  much  did  he  lose  ? 

11.  A  young  man,  having  $1240,  spent  6^  %  of  it  for  clothes 
and  board,  10  %  of  it  for  a  horse,  and  12^  %  of  it  in  travel- 
ing ;  how  much  had  he  left  ? 

12.  A  grocer  bought  450  pounds  of  coffee,  and  found  that 
10  %  of  it  was  damaged ;  how  much  of  it  was  good  ? 

13.  A  flock  of  175  sheep  increased  20  %,  how  large  was  it 
afterwards  ? 

CASE  H. 

Arti  132. — To  find  what  per  cent,  one  number  is  of 
another. 
EXAMPLE  14. — What  per  cent,  of  $10  is  $2.50  ? 

Process. — Since  $2.50  is  a  certain  per  cent,  of  jn\2  59 

$10,  the  same  per  cent,  of  SI.  is  A-  of  $2.50,  —sf 

which  is  $.25=25%.     Therefore  $2.50  is  25%  Ans.,  .25  =  25  %. 
of  $10. 

RULE. — Divide  the  number  which  is  the  percentage  by  the 
other  number. 

Eate  =  Percentage  -f-  Principal. 

15.  What  %  of  $75.  is  $5  ?  $10  ?  $20  ?  $25  ? 

16.  What  %  of  $87.50  is  $5.25  ?  $7.87^  ? 

17.  What  %  of  $60.  is  $6  ?  $9  ?  $12  ? 

18.  What  %  of  $56.  is  $7  ?  $14  ?  $21  ? 

19.  What  %  of  $150  is  $9  ?  $15  ?  $50  ? 

20.  What  %  of  $200  is  $8  ?  $16  ?  $24  ?  $50  ? 

21.  What  %  of  $1000  is  $50  ?  $60  ?  $75  ? 

22.  A  farmer  raised  500  bushels  of  wheat  and  kept  50  bushels 
for  family  use  ;  what  per  cent,  of  it  did  he  keep  ? 


PROMISCUOUS  EXAMPLES.  187 

23.  A  regiment  consisting  of  900  soldiers  lost  75  of  them  in 
a  battle  ;  what  per  cent,  was  it  ? 

24.  A  young  man  having  $350,  has  spent  850  of  it ;  what 
per  cent,  of  it  has  he  spent  ? 

25.  A  grocer  bought  560  gallons  of  molasses,  and  found  that 
56  gallons  had  leaked  out ;  what  was  the  percentage  ? 

26.  A  miller  took  15  bushels  of  corn  for  grinding  300  bush. ; 
what  per  cent,  was  it  ? 

CASE  IIL 

Art   ,  133, — To  find  the  principal  when  a  certain  per 
cent,  of  it  is  known. 

EXAMPLE  27. — A  man  gave  $50  to  benevolent  objects,  which 
was  10  %  of  his  income  ;  what  was  his  income  ? 

Process. — Since  $50  is  10  %  of  his  income,  his  income  $  cts. 

was  as  many  dollars  as  .10  is  contained  times  iu  50,      .  10)50 . 00 
which  is  500.     Therefore  his  income  was  $500.  .Ans.  $500. 

Or,  since  $50  is  ^  of  his  income,  the  whole  of  it  is 
$50X10=4500. 

RULE. — Divide  the  given  percentage  by  the  rote  per  cent. 
Principal—  percentage  -f-  rate. 

28.  $5  is  6  %  of  how  many  dollars  ? 

29.  12  pound  is  10  %  of  how  many  pounds  ? 

30.  15  gallons  is  25  %  of  how  many  gallons  ? 

31.  37%  bushels  is  18%  %  of  how  many  bushels  ? 

32.  $75  is  5  %  of  how  many  dollars  ? 

33.  A  man  pays  an  income  tax  of  4  %  amounting  to  $84  ; 
what  is  his  income  ? 

34.  A  debtor,  whose  property  is  worth  $4,500  is  able  to  pay 
only  75  %  of  what  he  owes ;  how  much  does  he  owe  ? 

35.  A  man  wishes  to  leave  his  daughter  an  income  of  $1,000 
a  year  ;  what  sum  must  he  invest  for  her  at  7  %. 

36.  A  miller  has  taken  1  barrel  of  flour  for  toll  at  2^  %\ 
how  many  barrels  of  flour  has  he  ground  ? 

37.  A  regiment  of  soldiers  lost  60  men  in  a  battle  which  was 
10  %  of  their  whole  number  ;  how  many  belonged  to  the  regi- 
ment before  the  battle  ? 


188  PROMISCUOUS  EXAMPLES. 

CASE  IV. 

Art.  134* — To  find  the  principal,  when  being  increased 
or  diminished  a  certain  per  cent.,  the  sum  or  remainder 
is  known. 

EXAMPLE  38. — A  man,  whose  property  has  increased  50  %  is 
now  worth  $15,000  ;  what  was  he  worth  before  ? 

Process. — Since  his  property  has  increased  50  %    $1.50)15000.00 
it  was  formerly  as  many  dollars  as  it  is  now  times         >*«<?   sinnnn 
$1.50,  L  e.  $10,000. 

Or  it  was  formerly  if£  or  f  of  $15,000=$10,000. 

Ex.  39. — A  man  who  has  lost  50  %  of  his  property,  is  now 
worth  $10,000  ;  what  was  he  worth  before  ? 

Process.  —Since  his  property  is  50  %  less  than  for-       . 50)10000.00 
merly,  it  was  then  as  many  dollars  as  it  is  now  times      £ns   $20000 

•ijp.OLI  it*    €m    ipiiUjUUiJ. 

Or  it  was  formerly  W  or  2  times  $10,000=420,000. 

BULE. — Divide  the  given  number  by  1,  with  the  rate  per 
cent,  added  or  subtracted  accordingly  as  the  percentage  has 
been  added  to  or  subtracted  from  the  required  number. 

Principal = principal  °r  percentage  -f-  by  1  or  rate. 

40.  A  farmer  has  raised  1,200  bushels  of  potatoes  this  year, 
which  is  25  %  more  than  he  raised  last  year;  how  many  did  he 
raise  last  year  ? 

41.  A  drover  has  purchased  800  sheep  which  is  20  %  less  than 
he  expected  to  purchase  ;  how  many  did  he  expect  to  pur- 
chase ? 

42.  My  tailor  told  me  it  would  require  6  yards  of  cloth  to 
make  me  a  suit  of  clothes.     Supposing  it  would  shrink  as  much 
as  5  %  of  its  length,  how  many  yards  should  I  have  purchased  ? 

43.  A  young  man  after  spending  18%  %  of  his  pocket  money 
had  $81.25  left ;  how  much  did  he  have  at  first  ? 

44.  A  lady  wishing  to  purchase  a  cloak  found  the  price  ($60) 
to  be  20  %  more  than  she  expected  to  pay  ;  how  much  did  she 
expect  to  pay  ? 

45.  A  lady  expecting  to  pay  $48  for  a  silk  dress  found  that 
to  be  20  %  more  than  the  price  ;  what  was  the  price  ? 


PKOMISCUOUS   EXAMPLES.  189 


Art.  135,—  Promiscuous  Examples  in  Percentage. 


46.  A  farmer  has  sold  118  tons  of  hay  this  year  which  is 
^more  than  he  sold  last  year,  and  33>^  %  less  than  he  expects 
to  sell  next  year  ;  how  much  did  he  sell  last  year  and  how  much 
does  he  expect  to  sell  next  year  ? 

47.  A  farmer  having  150  tons  of  hay  expects  to  sell  90  tons  ; 
what  per  cent,  is  that  of  the  whole  ? 

48.  A  farmer  has  sold  75  tons  of  hay  which  is  60  %  of  all  he 
had  ;  how  much  did  he  have  ? 

49.  A  liquor  dealer  bought  a  hogshead  of  rum  and  mixed 
33i/j  %  of  water  with  it  ;  how  much  did  it  make  ? 

50.  At  another  time  he  mixed  75  gals,  of  water  with  225  gals. 
of  brandy  ;  what  per  cent,  of  the  mixture  was  water  ? 

51.  At  another  time  he  filled  up  a  cask  containing  ale  with 
20^  gals,  of  water  which  was  37)^  %  of  what  the  cask  would 
contain  ;  how  much  did  the  cask  hold  ? 

52.  At  another  time  he  mixed  some  wine  with  15  %  of  water 
and  then  had  57^  gallons  ;  how  much  wine  was  there  ? 

53.  In  a  certain  town  the  population  is  2750,  and  4  %  are  col- 
ored ;  how  many  colored  persons  live  in  the  town. 

54.  In  another  town  there  are  2040  whites  and  360  blacks  ; 
what  per  cent  of  the  whole  population  are  black  ? 

55.  In  another  town  8%  %  of  the  population  are  colored,  of 
whom  there  are  136  ;  what  is  the  whole  population  ? 

56.  In  another  town  the  population  has  increased  in  10  years 
127)£  %,  and  is  now  2275  ;  what  was  it  ten  years  ago  ? 

57.  The  expenses  of  a  family  are  18%  %  greater  this  year 
than  the  last,  when  they  amounted  to  $963.  75  ;  how  much  are 
they  this  year  ? 

58.  A  debtor  whose  property  is  worth  $6,750,  is  able  to  pay 
only  67^  %  of  what  he  owes  ;  how  much  can  he  pay  ? 

59.  A  man  worth  $10,000  has  invested  $2,520  in  government 
bonds  ;  what  per  cent  of  his  property  is  thus  invested  ? 

60.  A  gentleman  traveling,  having  spent  75  %  of  his  money 
found  that  he  had  $75  left  ;  how  much  had  he  at  first  ? 


190  APPLICATIONS   OF  PERCENTAGE. 

Applications  of  Percentage. 

Art.  136. — Percentage  is  applicable  to  Commission  and 
Brokerage,  Stocks  and  Gold  at  a  Premium,  Insurance, 
Profit  and  Loss,  Interest,  Discount,  Taxes,  Duties,  Part- 
nership, Bankruptcy,  Exchange,  &c. 

Some  of  these  are  so  much  like  percentage  that  they  scarcely  need 
to  be  separately  treated  except  in  a  few  particulars. 

Art.  137.— Commission  is  the  percentage  paid  to  a 
commission  merchant  or  agent  doing  business  for  another. 
It  is  calculated  the  same  as  percentage. 

A  consignment  consists  of  goods  sent  to  a  person  to  sell 
on  commission.  The  gross  proceeds  are  the  whole  amount 
of  the  sales.  The  net  proceeds  are  what  is  left  after  de- 
ducting the  expenses. 

Art.  138,— An  Account  Of  Sales  is  a  written  statement 
of  goods  sold  on  commission,  with  the  prices,  gross  and 
net  proceeds,  &c. ;  as 

SALES  OF  PRODUCE  CONSIGNED  BY  THOS.  FAY  &  Co.,  DETEOIT. 

1867.  Sold  to  Produce.  Price. 


Aug  1 

Rogers  &  Son 

Flour,  20  bbls.  .  $11.  00 

$  cts. 
220  00 

"    8 
11  25 

C.  Jones  &  Co  
T.  Agnew  &  Co.  ... 

Wheat,  300  bu...  2.50 
Corn,  500  bu...  90 

750.00 
450.00 

Charges.  $1420.00 

Freight  on  20  bbls @  50  cts.  $10.00 

800  bu @10ctP-     80.00 

Cartage  and  Storage 12.50 

Commission  on  $1240  @  2)4  % 31. 95        124 . 45 

Net  proceeds TTT~       $1295.55 

SMITH,  MYGATT  A-  CO. 

New  York,  Aug.  31,  1867. 


BEOKEEAGE — STOCKS.  191 

Art.  139. — Brokerage  is  the  percentage  paid  to  brokers. 
It  is  sometimes  called  discount. 

A.  Broker  is  one  who  exchanges  or  loans  money,  buys 
and  sells  stocks,  also  goods  not  in  his  own  possession. 

Art.  140. — Stocks  are  money  or  property  invested  in. 
Banks,  incorporated  or  chartered  Companies,  Bonds,  &c. 
They  are  divided  into  shares,  usually  of  $100  each,  for 
which  Certificates  or  Scrip  is  issued,  liable  to  be  bought 
or  sold. 

When  stocks  sell  for  what  they  originally  cost,  they 
are  at  par  ;  when  they  sell  for  more,  they  are  above  par, 
at  a  premium  or  advance  ;  and  below  par,  or  at  a  discount, 
when  they  sell  for  less. 

The  premium  or  discount  is  a  certain  percentage  on 
the  par  value,  to  be  added  or  subtracted  from  it,  in  find- 
ing the  market  value. 

The  market  value  of  any  number  of  shares  is  found  by 
multiplying  it  by  the  market  value  of  a  single  share. 

Stocks  are  quoted  :  at  par,  100  ;  at  a  premium  of  1  %y 
101;  2J&  102|;  18|,  118|  ;  at  a  discount  of  5  %,  95  ; 
12*&87*;  25^,75,  &c. 

Stockholders  are  the  owners  of  stock. 

A  Dividend  is  what  is  paid  to  stockholders  as  their 
part  of  the  profit  or  gain. 

Bonds  are  securities  for  money  loaned,  bearing  inter- 
est, issued  by  Corporations  or  Governments. 

The  United  States  have  issued  the  following  bonds  : 

U.  S.  5's,  paying  5  %  interest  in  gold,  and  payable  in 
1871  and  1874. 

U.  S.  6's,  paying  6  %  interest  in  gold,  and  payable  in 
1867..  1868,  and  1881. 

U.  S.  5-20's,  paying  6  %  interest  in  gold,  and  payable 
in  5  to  20  years. 


192  APPLICATIONS   OF  PERCENTAGE. 

U.  S.  10-40's,  paying  5  %  interest  in  gold,  and  payable 
in  10-40  years. 

U.  S.  7-30's,  paying  7^,  or  7.30%  interest,  in  currency, 
and  payable  in  three  years  from  their  date. 

Art.  141. — Gold9  at  a  premium,  is  bought  and  sold  the 
same  as  stocks. 

Art.  142. — Insurance  is  security  against  loss. 

Fire  Insurance  is  security  against  loss  by  fire  ;  Marine 
Insurance,  against  loss  on  the  ocean,  &c.  Insurance,  also, 
secures  a  certain  allowance  in  case  of  accident,  sickness,  or 
death.  The  last  is  called  Life  Insurance. 

The  Policy  is  the  written  contract. 

The  Premium  is  a  certain  percentage  on  the  amount 
insured. 

Art.  143.  —  Promiscuous  Examples  in  Commission, 
Brokerage,  Slocks,  Gold,  and  Insurance. 

[These  examples  are  to  be  done  the  same  as  others  in 
Percentage.] 

The  amount  bought  or  sold,  collected,  invested  or  in- 
sured, or  the  par  value  of  stocks  and  gold,  is  the  principal ; 
to  be  found,  if  required,  by  Case  III.  in  Percentage  ;  or 
Case  IV.  when  the  percentage  is  to  be  deducted  from 
the  given  sum,  or  has  been  deducted  from  the  required 
sum. 

The  per  cent,  is  the  rate ;  to  be  found,  if  required,  by 
Case  n.  in  Percentage. 

The  commission,  brokerage,  dividend,  premium,  or  dis- 
count, is  the  percentage ;  to  be  found,  if  required,  by  Case 
I.  in  Percentage. 

Ex.  1.  What  is  the  commission  for  selling  goods  amounting 
to  $2500,  at  3^  per  cent.  ?  * 

Process,  the  same  as  in  Percentage,  Case  L 


PKOMISCUOUS  EXAMPLES.  193 

2.  A  commission  merchant  received  $87.50  for  selling  goods 
amounting  to  $2500 ;  what  per  cent,  was  his  commission  ? 

Process. — Percentage,  Case  II. 

3.  A  commission  merchant  received  $87.50  for  selling  goods 
at  3-£  per  cent. ;  what  was  the  amount  he  sold  ? 

Process. — Percentage,  Case  HI. 

4.  A  commission  merchant  received  $2587.50  for  the  pur- 
chase of  goods,   after  deducting  3-^  per  cent,   commission ; 
what  was  the  amount  of  the  goods  he  purchased  ? 

Process. — Percentage,  Case  IV.;  commission  to  be  deducted. 

5.  A  commission  merchant,  after  deducting  3£  %  commis- 
sion from  the  whole  sum  he  had  received,  had  a  balance  of 
$2500  for  the  purchase  of  goods  ;  what  was  the  whole  sum  he 
received  ? 

Process. — Percentage,  Case  IV. ;  commission  already  deducted. 

6.  A  broker  in  New  York   exchanged    $1500,   uncurrent 
money,  at  ^  per  cent,  discount ;  how  much  was  his  brokerage  ? 

7.  A  broker  has  $5012^  to  invest  in  bank  stock,  after  de- 
ducting \  %  for  brokerage  ;  how  much  is  to  be  invested  ? 

8.  A  merchant  gave  a  broker  $1000  uncurrent  money,  and 
received  from  him  $990  current  money  ;  what  per  cent,  was 
the  brokerage  ? 

9.  A  merchant  gave  a  broker  $10  for  exchanging  some  un- 
current money,  at  %  %  ;  what  was  the  amount  ? 

10.  A  broker,  after  deducting  %  %  for  brokerage,  paid  back 
$1985  current  money ;  how  much  uncurrent  money  had  he 
received  ? 

11.  What  is  the  value  of  12  shares  of  railroad  stock,  at  a 
premium  of  5  %  ? 

12.  What  is  the  value  of  the  same  at  5  %  discount  ? 

13.  How  much  stock,  at  5  %  premium,  can  be  bought  for 
$5250  ? 

14.  How  much  stock,  at  5%  discount,  can  be  bought  for 
$4750? 


194  APPLICATIONS   OF   PERCENTAGE. 

15.  When  gold  is  125,  how  much  is  $500  in  gold  worth  in 
paper  currency? 

16.  When  gold  is  125,  how  much  is  $500  in  paper  currency 
worth  in  gold  ? 

17.  What  must  be  paid  for  insuring  a  house  valued  at  $2500, 
at  1}£  %  premium  ? 

18.  Paid  $31.25  for  insuring  a  house  valued  at  $2500  ;  what 
per  cent,  was  the  premium  ? 

19.  Paid  $37.00  for  insuring  a  house,  at  1  %  premium;  what 
was  the  amount  of  the  insurance  ? 

20.  For  how  much  must  a  house,  valued  at  $2500,  be  in- 
sured, at  \%  %,  to  cover  both  its  loss  by  fire  and  the  cost  of 
the  insurance  ? 

21.  What  is  the  commission  for  selling  goods  amounting  to 
$6500,  at  3  %  ? 

22.  What  is  the  brokerage  on  $1500  uncurrent  money,  at 

#x? 

23.  An  agent  has  received  $1230  for  the  purchase  of  goods, 
after  deducting  2)^  %  commission  ;  what  will  be  the  amount 
of  the  goods,  and  what  his  commission  ? 

24.  How  much  money  at  1  %  discount  will  pay  a  note  for 
$1500  ? 

25.  What  is  the  value  of  10  shares  of  bank  stock  at  a  pre- 
mium of  12  %  ? 

26.  How  many  shares  of  railroad  stock  10  %  below  par  will 
pay  a  debt  of  $1800  ? 

27.  What  must  be  paid  for  the  insurance  on  a  store  and 
goods  valued  at  $18000,  at  3  per  cent  ? 

28.  How  much  current  money  should  a  broker  give  in  ex- 
change for  $560,  at  yz  per  cent,  discount  ? 

29.  What  will  be  the  amount  of  a  bill  of  goods  to  be  pur- 
chased with  the  balance  of  $5616,   received  by  remittance, 
after  deducting  4  %  commission  ? 

30.  How  much  are  15  shares  in  an  Insurance  Company 
worth  at  6^  %  advance  ? 

31.  What  is  the  premium  on  an  insurance  amounting  to 
$5620,  at 


PROMISCUOUS    EXAMPLES.  195 

32.  When  gold  is  118%,  how  much  in  currency  should  be 
received  for  $60  interest  on  a  Government  bond,  payable  in 
gold? 

33.  An  agent  in  Chicago  has  bought  grain  for  a  flouring 
mill  in  New  York  amounting  to  $3500 ;  what  is  his  commis- 
sion, at  3^  per  cent.  ? 

34.  What  should  a  broker  give  in  exchange  for  $640  uncur- 
rent  money,  at  3%  %  discount  ? 

35.  A  Southern  merchant  has  remitted  to  his  agent  in  Phil- 
adelphia $2370}^,  for  the  purchase  of  goods,  after  deducting 
his  commission,   at  3^  % ;  what  will  be  the  amount  of  the 
goods  to  be  purchased  ? 

36.  What  should  a  broker  deduct  from  the  amount  of  a 
draft  for  $850,  taken  in  exchange  for  currency,  at  %  %  dis.  ? 

37.  What  are  18  shares  of  bank  stock  worth  at  16%  %  adv.  ? 

38.  What  is  the  premium  on  the  insurance  of  a  house  val- 
ued at  $2800,  at  4>£  %  ? 

39.  When  gold  is  125,  how  much  of  it  will  pay  a  debt  of 
$600? 

40.  A  gentleman  wishes  a  horse-dealer  to  purchase  for  him 
a  span  of  horses  worth  $900,  and  is  willing  to  allow  him  3%  % 
commission ;  what  will  the  horses  cost  him  ? 

41.  A  gentleman  gave  a  horse-dealer  $806  for  a  span  of 
horses  and  his  commission  at  4  % ;  what  was  paid  for  the 
horses  alone  ? 

42.  What  is  a  draft  for  $480  worth  at  %  %  discount  ? 

43.  What  are  6  shares  in  a  steamboat  company  worth  at  a 
premium  of  100  per  cent.  ? 

44.  When  gold  is  1.27%,  how  much  currency  will  pay  the 
duties  (payable  in  gold)  on  a  bill  of  imported  goods  amount- 
ing to  $3750.50  ? 

45.  What  must  be  paid  for  insuring  a  steamboat  valued  at 
$75000,  at  6>£  per  cent.  ? 

46.  The  insurance  of  a  steamship,  valued  at  $250000,  costs 
$10000  ;  what  per  cent,  is  the  premium  ? 

47.  A  merchant  in  San  Francisco  has  remitted  to  his  agent 
im  Hew  York  $4888  in  gold  for  the  purchase  of  goods,  after 


196  PROFIT  AND  LOSS. 

deducting  4  %  commission  ;  gold  at  the  time  being  125  ;  what 
is  the  commission  ? 

48.  What  are  10  shares  of  the  Corn  Exchange  Bank  worth 
at  a  premium  of  16%  %  ? 

49.  Bought  a  house  lot  for  $10000  when  gold  was  at  par ; 
what  ought  it  to  be  worth,  gold  being  $130  ? 

50.  A  merchant  had  paid  3}^  %  for  insurance  on  his  stock 
of  goods  at  $15000,  for  10  years.     After  that  time  they  were 
destroyed  by  fire  ;  what  did  he  gain  by  having  them  insured  ? 


PROFIT  AND  LOSS. 

Art.  144. — Profit  (or  Gain)  and  Loss  are  usually  esti- 
mated at  a  certain  percentage  on  the  cost  (principal.) 

The  profit  or  loss,  at  any  rate  per  cent,  is  found  by 
Case  I.  in  Percentage. 

The  per  cent,  of  profit  or  loss  by  Case  II. 

The  cost  by  Case  III.  or  IV. 

The  profit  or  loss  =  cost  X  Per  cent. 

The  per  cent,  of  profit  or  loss  ==  the  profit  or  loss  -r- 
cost. 


-I 


.  the  profit  or  loss  -r-  per  cent,  or 

The  COSt  =  J  %.  gain 

the  selling  price  -r- 1  or  per  cent,  or 


The  profit  or  loss  is  the  difference  between  the  cost 
and  selling  price. 

The  selling  price  is  the  cost  with  the  profit  added  or 
the  loss  subtracted. 

When  the  rate  per  cent,  is  an  aliquot  part  of  100,  it  is 
often  more  convenient  to  express  it  in  that  form  ;  as 
25  =  J 


EXAMPLES. 

1.  When  cloth  costs  $4  and  is  sold  at  25  %  profit,  what  is 
the  profit  and  selling  price  ? 


EXAMPLES.  197 

Process  as  in  Percentage,  Case  I.  S4X-25,  or  £  =  $1.00  profit, 
which  added  to  the  cost  is  *5.00  the  selling  price. 

2.  When  the  cost  of  cloth  is  $4  a  yard  and  the  selling  price 
$5.00,  what  per  cent,  is  the  profit  ? 

Process. — 55 — 84=  SI.  (gain)  then  as  in  Percentage,  Case  IE. 
$14- $4  =  .25,  or  25%. 

3.  When  cloth  is  sold  at  25  %  profit  and  the  profit  is  Si  a 
yard,  what  did  it  cost  ? 

Process  as  in  Percentage,  Case  m.  Sl.00-j-.25  or  ±=  S4.00  (the 
cost. ) 

4.  When  cloth  is  sold  for  $5.00  a  yard  and  25%  is  thus 
gained,  what  did  it  cost  ? 

Process  as  in  Percentage,  Case  IV.  $5. 00 -j- $1.25  =  $4.00  the 
cost. 

The  prices  in  the  following  examples  may  be  considered  as  the 
prices  per  yard,  pound,  &c.,  &c. 

5.  Cost  10  cents  ;  profit  20  %  ;  selling  price  is  what  ? 

6.  Cost  15  cents  ;  selling  price  18  ;  gain  per  cent,  is  what  ? 

7.  Profit  5  cents  ;  per  cent.  20  ;  cost  is  what  ? 

8.  Selling  price  20  cents  ;  loss  25  %  ;  cost  is  what  ? 

9.  Cost  $1. ;  selling  price  Si.  25  ;  gain  per  cent,  is  what  ? 

10.  Profit  20  cents  ;  per  cent.  15  ;  cost  is  what  ? 

11.  Selling  price  35  cents  ;  profit  16%  %  ;  cost  is  what  ? 

12.  Cost  32  cents  ;  profit  12)£  %  ;  selling  price  is  what  ? 

13.  Cost  $3;  profit  50  cents  ;  per  cent,  is  what  ? 

14.  Cost  $300;  loss  $75  ;  per  cent,  is  what  ? 

15.  Selling  price  $75;  loss  25  %  ;  cost  is  what  ? 

16.  Profit  $625;  per  cent.  12}£  ;  cost  is  what  ? 

17.  Cost  $150;  profit  16%  %  ;  profit  is  what  ? 

18.  Cost  $5.20  ;  profit  10  % ;  selling  price  is  what  ? 

19.  Selling  price  $3612>£  ;  loss  §%  %  ;  cost  is  what  ? 

20.  Cost  $875;  selling  price  $1050  ;  per  cent,  is  what  ? 

21.  Cost  $1000  ;  profit  $70;  gain  per  cent,  is  what  ? 

22.  Cost  $1000  ;  loss  7  %  ;  selling  price  is  what  ? 

23.  Loss  $120;  per  cent.  6  ;  cost  is  what  ? 

24.  Profit  $300;  per  cent.  6  ;  cost  is  what  ? 

25.  Selling  price  6^  cents  ;  cost  6  cents  ;  per  cent,  is  what  ? 


198  INTEREST. 

26.  Selling  price  $5.25  ;  cost  $6;  loss  per  cent,  is  what  ? 

27.  Cost  $2000  ;  profit  18%  % ;  selling  price  is  what  ? 

28.  Bought  a  horse  for  $175  and  sold  him  for  $210  ;  what 
was  the  profit  per  cent.  ? 

29.  Sold  a  house  for  $2850,  5  %  less  than  cost ;  what  did  it 
cost? 

30.  Bought  a  piece  of  land  for  $1500  and  sold  at  an  ad- 
vance of  33}^  %  ;  what  was  the  selling  price  ? 

31.  Gained  in  trade  $1600,  which  16%  %  of  the  capital  en> 
ployed  ;  what  was  the  capital  ? 

32.  A  wool  merchant  bought  36000  Ibs.  of  wool  at  56  cents  a 
pound,  the  expenses  on  it  were  $108,  and  he  sold  it  for  62  y2 
cts.  a  pound  ;  how  much  per  cent,  did  he  gain  ? 

33.  If  16  cwt.  31  Ibs.  of  sugar  cost  $203.875,  for  what  must 
it  be  sold  per  pound  to  gain  24  %  ? 

34.  A  grocer  sold  tea  for  $.75  a  pound,  and  lost  12 %  %  ; 
what  did  it  cost  ? 

35.  A  grocer  sold  tea  for  $1  a  pound,  and  gained  33  V^  %  ; 
what  did  it  cost  ? 

36.  A  grocer  gained  12}£  cts.  a  pound  on  tea,  which  was 
25  %  ;  what  did  the  tea  cost  per  pound  ? 


INTEREST. 

Art.  145. — Interest  is  a  certain  percentage  paid  for  the 
use  of  money  for  a  specified  time. 

The  principal  is  the  money  lent,  or  loaned. 

The  rate  is  the  per  cent,  paid  annually  or  per  annum. 

The  amount  is  the  principal  with  the  interest  added  to  it. 

Simple  interest  is  the  interest  on  the  principal  only. 
Compound  interest  is  on  the  principal  and  interest  al- 
ready due.  When  only  the  word  interest,  is  used,  simple 
interest  is  meant. 

Legal  interest  is  at  the  rate  fixed  by  law.  In  all  the 
United  States  it  is  6  per  cent,  except  New  York,  Michigan, 


INTEREST. 


199 


Wisconsin,  Minnesota,  S.  Carolina,  and  Georgia,  7  % ; 
Louisiana,  5  % ;  Florida,  Alabama,  Mississippi,  and  Texas, 
8  %  ;  Kansas  and  California,  10  %  ;  Oregon,  12^  %. 

EXAMPLE  1.— What  is  the  interest  of  $125.50  for  2  years  5 
months  and  21  days,  at  6  %  ? 

Process. — Since  the  interest  of  $1.00  for  a  year  is  $.06  (6  cts.)  the 
interest  of  $125.50  for  1  year  is  (.06  times  $125.50  or  125.50  times 
$.06)  $7.53  ;  for2  years  ($7.53X2)  $15.06  ;  for  5  months  (fa  of  $7.53) 
$3.138  and  for  21  days  (ffc  or  -fa  of  $.628  the  interest  for  1  month  or 
-iV  of  a  year)  $.439.  Therefore,  the  interest  for  the  whole  time  is 
$18.637. 

It  is  sufficiently  accurate  to  count  £  mill  or  more  as  1  mill,  and  re- 
ject less  fractions. 

$125.50 
.06 

7.53    =1  year. 
2 

15.06    =2  years. 
^  year's  2 . 01    =4  months. 

1  4  month's       .  628  =1  month. 
£  month's          .314  =15  days. 
^15  days  .105  =5  days. 

15  days  .021  =1  day. 

$18.638  whole  time. 


A 


$125.50 
.06 
7T53~ 

2 

15.06 
3.138 
.439 

AM.     $18.637 


int.  1  year. 

"  2  years. 
;  5  months. 
21  days. 

whole  time. 


RULE. — To  find  the  interest  for  one  year,  multiply  the  prin- 
cipal by  the  rate. 

For  two  or  more  years,  multiply  this  product  by  the  num- 
ber of  years. 

For  months  take  aliquot  parts  of  a  year's  interest. 

For  days  take  aliquot  parts  of  a  month's  interest. 

The  above  rule  is  applicable  to  any  rate  ;  but  at  6  % 
the  following  method  is  preferable  : 

Art,  146.— At  6  %  the  interest  of  $1  being  $.06  a  year 
or  12  months,  it  is  half  as  many  cents  as  months.  Hence 
the  interest  of  $1  for 


1  month $.005 

2  months $.01 

3  months $.015 


6  months $.03 

9  months $.045 

11  months $.055 


200  INTEREST. 

Also,  the  interest  of  $1  for  one  month  (30  days)  being 
$.005,  it  is  1  mill  for  every  6  days,  or  £  as  many  mills  as 
days. 

EXAMPLE  1.— By  second  method. 


Process.  —  The  interes 

2  years=  
5  montlis= 
21  days  =  

whole  time=  .  .  . 

;  of  $1.  for 

$.12 
$.025 
$.0035     Therefore 

\              $125.50 
.1485 

62750 
100400 
50200 
12550 

$.1485 

6)18.  636750  at  6  % 

3.  106125  at  \% 

I        $21. 742875  at  7% 
The  interest  at  6  %  divided  by  6  is  the  interest  at  1  %,  and  this 
multiplied  by  7  is  the  interest  at  7  %  ;  or  if  multiplied  by  any  other 
rate  it  would  be  the  interest  at  that  rate. 

EULE. — Find  the  interest  of  $l/or  the  given  time,  allowing 
6  cents  for  every  year,  half  as  many  cents  as  months  and  as 
many  mills  as  there,  are  times  6  days ;  then  multiply  it  by  the 
given  number  of  dollars. 

EXAMPLES. 

In  the  following  examples,  pupils  should  use  both  methods  of  cal- 
culating interest  till  they  become  familiar  with  them,  then  either  as 
they  may  prefer,  or  the  teacher  may  direct.  There  may  be  a  slight 
difference  in  the  answers  obtained  by  the  different  methods.  In 
business,  interest  is  more  generally  computed  by  tables. 

What  is  the  interest  of  $1  for 

(1.)  1  year  5  months  18  days,  at  6%  ?  also  of  $125.00  ? 

(2.)  2     "    4        "      12    "  "  6%?  "  875.50? 

(3.)  3     "    7        "      15    "  "6,%?  "  $210.25? 

(4)  4    "    8        "      20    "  "  6%?  "  $87.625? 

(5.)  5     "    9        "      21    "  "6)£%?  "  $10.12',? 

(6.)  6     "1        "        3    "  "6%?  "  $256.00? 

(7.)  7     "0        "        4    "  "6%?  "  $160.50? 

(8.)  9        "      25    "  "  6%?  "  $62.375? 

(9.)  10  "10        "      10    "  "6%?  "  $100.00? 

(10.)  27    "  "  6%?  "  $1000.00? 


INTEKEST.      ,  201 

What  is  the  interest  of 

(11. )  145. 00  for  1  year  3  months  6  days,  at  7  %  ? 
(12.)  $130.00  "  2  "  1  "  3  "  7%? 
(13.)  $234.50  "  3  "  8  "25  "  7  %  ? 
(14.)  $15.87K  "  4  "  5  "  15  "  7 %  ? 
(15.)  $112.5614  "  5  "10  "  *9  "  4)£X? 
(16.)  $215.00  "3  "11  "  13  "  5%? 
(17.)  $321.18%  "  2  "  6  "  17  "  7^? 
(18.)  $400.  "  4  "  "10  "  S%? 

(19.)    $500.          "   1     "    4       "22       "       5%? 
(20.)    $650.          "  3     "    3       "       3       " 
(21.)      $36.50      "  6       "  at 

(22.)    $118.75      "  19       "       7%? 

(23.)  $90.00  "  2  "  1  "  6  "  4>£%? 
(24.)  $124.80  "  3  "  5  "  7  "  5%? 
(25.)  $84.30  "  1  "  6  "  12  "  6%? 
(26.)  $100.  "  4  "  7  "  18  «•  7%? 
(27.)  $72.70  "  2  "  2  "  2  "  7  %  ? 
(28.)  $95.50  "  1  "  1  "  1  "  6%? 
(29.)  $100.  "  4  "  8  "  7%? 

{30.)  $1000.          "  29       "       ±Yz%e! 

31.  What  is  the  amount  of  $10  for  1  year  6  months  at  6  %  ? 

32.  What  is  the  amount  of  $25  for  6  months  12  days  at  7  %  ? 

33.  What  is  the  amount  of  $50  for  2  years  20  days  at  7  %  ? 

34.  What  is  the  amount  of  $60  for  2  months  15  days  at  6  %  ? 

35.  What  is  the  amount  of  $75  for  1  year  10  days  at  7  %  ? 

36.  What  is  the  amount  of  $100  for  7  months  18  days  at  5  %? 

37.  What  is  the  interest  of  $120  for  2  years  9  months  at  6  %  ? 

38.  What  is  the  amount  of  $150  for  3  months  3  days  at  7  %  ? 

39.  What  is  the  amount  of  $1  for  100  years  at  7%  ? 

40.  What  is  the  amount  of  $10  for  10  days  at  6  %  ? 

41.  What  is  the  interest  of  $125.50  from  July  1,  1858,  to 
Jan.  1,  1859,  at6%? 

42.  What  is  the  amount  of  $200  from  Aug.  1,  1860,  to  Nov. 
16,  1860,  at  7%  7 

43.  What  is  the  amount  of  $137.50  from  May  16,  1865,  to 
January  1,  1866,  at  6  %  ? 

9* 


202  INTEREST. 

44.  What  is  the  interest  of  $300  from  June  1,  1864,  to  Feb. 
10,  1865,  at  7  %? 

45.  What  is  the  amount  of  $250  from  July  10,  1865,   to 
March  1,  1866,  at  6,%? 

46.  What  is  the  amount  of  $320  from  Sept.  15,  1867,  to 
April  1,  1868,  at  5%?' 

47.  What  is  the  interest  of  $250  from  July  15,  1866,  to  Jan. 
1,  1867,  at  6%? 

48.  What  is  the  amount  of  $560  from  January  1,  1867,  to 
June  16,  1867,  at7%? 

49.  What  is  the  amount  of  $67.50  from  July  21,   1866,  to 
Sept.  1,  1866,  at  6  %  ? 

50.  What  is  the  amount  of  $100  from  June  15  to  July  1,  at 
5>0"? 

Exact  Interest. 

Art.  147. — In  the  preceding  examples  30  days  have 
been  allowed  to  each  month,  which  makes  a  year  to  con- 
sist of  360  instead  of  365  days.  To  find  the  exact  inter- 
est we  must  count  the  exact  number  of  days  in  each 
month  (see  Table  of  Time,  Art.  50,)  and  consider  *each 
day  ^5  part  of  1  year. 

EXAMPLE  51. — What  is  the  exact  interest  of  $36  from  March 
1  to  June  9,  at6%? 

$36 
n/» 

Process. — The  number  of  days  are  31  in  March, 
30  in  April,  31  in  May,  and  8  in  June  (31+30+ 
31+8,)  100.     And  $2.16,  the  interest  for  a  year, 
divided  by  365,    gives  the  interest  for  1   day,     365)216. 00($.  59+ 
which  multiplied  by  100,  gives  the  interest  for  1825 

100  days.     For  convenience  we  multiply  by  100  3350 

first,  and  then  divide  by  365.  3295 

55 

52.  What  is  the  exact  interest  of  $50  from  July  1  to  Sept. 
16,  at  6  %  ? 

53.  What  is  the  exact  interest  of  $1000  from  Oct.  10  to  Jan. 
1,  at  7^? 


PARTIAL   PAYMENTS.  203 

54.  What  is  the  exact  interest  of  $100  from  July  1  to  Aug. 
l,at6/V? 

55.  What  is  the  exact  interest  of   $100  from  Feb.   10  to 
March  10,  at  6  %  ?  if  it  is  leap-year  ? 

Art.  148. — Interest  of  Sterling  Money. 
EXAMPLE  56.— What  is  the  interest  of  £50  10s.   6d.    for  1 

year  4  months,  at  5  per  cent.  ? 

£50,525 
5 


3)2.52625 
84208 

Process. ~£50  10s.  6d.=  (Article  121)  £50.525,    on       £3  36833 
which  the  interest  4  m.  =%  year,  found  as  in  Federal 
Money,  is  £3.36833,  or  (Art.  122)  £3  7s.  4d.  1.5968far. 


1.59680 
RULE. — Reduce  the  shillings,  &c.,  to  the  decimal  of  pound. 

Then  proceed  "as  in  Federal  Money,  and  reduce  the  decimals 

to  shillings,  &c. 

What  is  the  interest  of 

Ex.  57.  £325  12s.  3d.  for  5  years,  at  6,%"? 

58.  £174  10s.  6d.  for  3  years  6  months,  at  6  %  ? 

59.  £150  16s.  8d.  for  4  years  7  months,  at  6  %  ? 

60.  £45  10s.  for  2  years,  at  4  %  ? 


PARTIAL  PAYMENTS, 

Art.  149. — A  partial  payment  is  the  payment  of  part 
of  a  note  or  debt  bearing  interest. 

A  Note  is  a  written  promise  to  pay  a  debt ;  as 

NEW  YOEK,  Sept.  1,  1867. 

Thirty  days  after  date  I  promise  to  pay  J.  V.  Peck, 
or  order,  three  hundred  and  twenty-five  -ffc  dollars,  for  value 
received.  SAM'L  A.  EOGEES. 


204  PARTIAL   PAYMENTS. 

Sam'l  A.  Rogers  is  the  drawer  or  maker. 
J.  V.  Peck  is  the  payee. 
$325.38  is  the  face  of  the  note. 

UNITED   STATES   RULE. 

When  partial  payments  have  been  made,  apply  the  payment^ 
in  the  first  place,  to  the  discharge  of  the  interest  then  due. 

If  the  payment  exceeds  the  interest,  the  surplus  goes  toward 
discharging  the  principal,  and  the  subsequent  interest  is  to  be 
computed  on  the  balance  of  principal  remaining  due. 

If  the  payment  is  less  than  the  interest,  the  surplus  of  the 
interest  must  not  be  taken  to  augment  the  principal,  but  interest 
continues  on  the  former  principal,  until  the  period  when  the 
payments  taken  together  exceed  the  interest  due,  and  then  the 
surplus  is  to  be  applied  towards  discharging  the  principal, 
and  interest  is  to  be  computed  on  the  balance  as  aforesaid. 

EXAMPLE  1. —  HARTFORD,  Jan.  1,  1860. 

For  value  received,  I  promise  to  pay  — Five  Hun- 
dred Dollars  on  demand,  with  interest  at  6  per  cent. 

Indorsements  :— May  1,  1861,  $175.     Sept.  16,   1862,  $25. 
Jan.  1,  1864,  $100.     July  25,  1864,  $120. 
What  was  due  Jan.  1,  1865. 


DATES. 


Note I860..  1  1 

1st  Payment,  1861 . .  5  1 

2d  Payment,  1862 .  .  9  16 

3d  Payment,  1864.  .   1  1 

4th Payment,  1864. .  7  25 


TIME  BETWEEN  DATES. 

Years.  Months.  Days. 
1  4 

1            4  15 

1            3  15 

6  24 

5  6 


Int.  of  $1, 
.08 
.0825 
.0775 
.034 
.026 


Settlement,..  1865..  1        1 

Principal $500. 

Interest  to  May  1,  1861  40. 

Amount 540. 

1st  Payment 175. 

Balance  and  new  principal  May  1,  1861 365. 

Interest  to  Sept.  16,  1862 $30. 11 

2d  Payment  (less  than  interest) 25.00 

Surplus  interest 5.11 


PABTIAL  PAYMENTS.  205 

Interest  of  the  same  principal  from  Sept.  16, 

1862  to  Jan.  1,  1864 28.29  33.40 

Amount 398.40 

3d  payment  (to  be  deducted)  100.00 

Balance  and  new  principal  298.40 

Interest  to  July  25,  1864 10.146 

Amount 308.546 

4th.Payment  (to  be  deducted) 120.000 

Balance  and  new  principal 188.546 

Interest  to  Jan.  1,  1865 4.902 

Amount  due  Jan.  1,  1865 $193.448 

Ex.  2.  A  note  of  $450  is  dated  June  16,  1860.     Interest  at 
7  per  cent. 

Indorsements  :— Aug.  1,  1861,  $20.     Jan.  13,  1863,  $220. 

"What  was  due  May  16,  1864  ? 

Ex.  3.  A  note  of  $300  is  dated  July  1,  1861.     Interest  at  6 
per  cent. 

Indorsements  :— Jan.  1,  18£3,  $15.     July  1,  1865,  $150. 

What  was  due  May  1,  1866  ? 

Ex.  4.  A  note  of  $620  is  dated  Sept.  9,  1863.     Interest  5 
per  cent. 

Indorsements  :— Dec.  21,  1863,  $75.     Sept.   8,  1864,  $200. 
June  20,  1865,  $20. 

What  was  due  Sept.  20,  1865. 

Ex.  5.  A  note  of  $750  is  dated  May  1, 1862.    Int'rst  6  per  cent. 

Indorsements  :— Nov.  16,  1863,  $50.     Sept.  1,  1864,  $175. 

What  was  due  May  1,  1865  ? 

Ex.  6.  A  note  of  $500  is  dated  Sept.  7,  1860.     Interest  7 
per  cent. 

Indorsements  : — Jan.   1,   1861,  $100.     June  19,   1861,  $10. 
Jan.  1,  1862,  $200. 

What  was  due  Sept  1,  1862  ? 

Ex.  7.  $600.  NEW  HAVEN,  Jan.  1,  1863. 

On  demand  I  promise  to  pay  D P &  Co. , 

or  order,  six  hundred  dollars,  for  value  received,  with  interest. 

P W . 

Indorsements  :— Jan.  1,  1864,  $100.     July  1,  $10. 

What  was  due  Jan.  1,  1865  ? 


206  PARTIAL  PAYMENTS. 

Ex.  8.  $1*00.  NEW  YOBK,  Dec.  25,  1863. 

On  demand,  we  promise  to  pay  G M . 

&  Co. ,  or  order,  fifteen  hundred  dollars,  i'or  value  received, 
with  interest. 

P W &  Co. 

Indorsements  : — Jan.  7,  1865,  $200.    July  7,  1865,  $25. 
What  was  due  Aug.  1,  1865  ? 

MERCANTILE  RULE. 

Art.  150. — Find  the  amount  of  the  principal  for  one  year, 
and  from  it  subtract  the  amount  of  each  payment  during  the 
year,  to  the  end  ofti;  the  remainder  mil  be  a  new  principal, 
with  which  proceed  as  before. 

If  the  time  of  settlement  is  less  than  a  year,  find  the  amount 
of  the  principal  for  such  a  portion  of  a  year,  and  subtract 
from  it  the  amount  of  the  payments  to  the  same  date. 

EXAMPLE  9. — A  note  of  $1000  is  dated  Jan.  1,  1864  interest 
6  per  cent. 

Indorsements :— March  12,  1864,  $200.  October  25,  $350. 
April  18,1865,  $100. 

What  was  due  June  18,  1865  ? 

Principal $1000. 

Interest  for  1  year 60. 

Amount 1060. 

1st  Payment  (March  10) $200.00 

Interest  till  Jan.  1,  1865 9.60 

2d  Payment  (Oct.  25) 350.00 

Interest  till  Jan.  1,  1865 3.85    563.45 

Balance  for  new  principal 496.55 

Interest  till  settlement 13.82 

Amount 510.37 

3d  Payment  (April  18,  1865) 100.00 

Interest  tiU  settlement 1.00    101.00 

Balance  due  June  18,  1865 $409.37 

Ex.  10.  A  note  of  $1000  is  dated  Jan.  1,  1866.  Interest  6 
per  cent. 


CONNECTICUT  RULE.  207 

Indorsements :— March  1,  $100.  May  25,  $100.  Sept.  1, 
$100.  July  1,  1867,  $500. 

What  was  due  Sept.  1,  1867  ? 

Ex.  11.  A  note  of  $500  is  dated  July  1,  1865.  Interest  7 
per  cent. 

Indorsements  :— Jan.  1,  1866,  $100.  April  1,  $100.  May 
16,  $100. 

What  was  due  July  1,  1866  ? 

Ex.  12.  A  note  of  $800  is  dated  May  1,  1866.  Interest  G 
per  cent. 

Indorsements:— Sept.  1,  $200.  Jan.  1,  1867,  $200.  March 
1,  $200. 

What  was  due  May  1,  1867  ? 

CONNECTICUT  RULE. 

Art.  151. —  The  Connecticut  rule  is  the  same  as  the  Mer- 
cantile when  a  payment  is  made  in  less  than  a  year;  other- 
wise it  is  the  same  as  the  U.  S.  Rule. 

Find  by  the  Connecticut  Rule  what  was  due  in  Example  1, 
page  204. 

Process. — The  same  as  by  U.  S.  Bule  till  the  third  payment  has 
been  deducted,  leaving 

Balance  for  a  new  principal $298.40 

Interest  for  1  year 17.90 

Amount  (Jan.  1,  1865) 316.30 

4th  Payment $120.00 

Interest  to  Jan.  1,  1865 3.12    123.12 

Amount  due  Jan.  1,  1865 $193.18 

By  the  same  rule  find  the  amounts  due  in  Examples  2,  3,  4, 
&c. 

If  the  answers  are  very  nearly  the  same  as>by  the  U.  S.  Rule 
they  may  be  considered  correct. 

RATE. 

Art.  152. — To  find  the  rate  when  the  principal,  interest,  or 
amount  and  time  are  given. 


208  INTEREST. 

EXAMPLE  1.— At  what  rate  will  $200  yield  $30  interest  in  2 
years  6  months  ? 

$200 
.01 


Process.— At  1%  the  interest  of  $200  for  6  mos.  £)2.00  1  year. 

2  years  6  months  is  $5 .    Therefore  if  the  in-  2 

terest  is  $30  the  rate  must  be  as  many  times  IjToO  2  yrs. 

1  %  as  $5  is  contained  times  in  $30,  or  6  %.  I.'OO  8  mos. 

"5^0  30.00 

Ans.    '      6% 

RULE. — Divide  the  given  interest  by  the  interest  of  the 
principal  at  1  %. 

Ex.  2.  At  what  rate  will  $320  yield  $72.80  in  3  yrs.  3  mos.? 

TIME. 

Artt  153* — To  find  tine  time  when  the  principal,  interest,  or 
amount  and  rate  are  given. 

Ex.  3.  In  what  time  will  $200  yield  $33  interest,  at  6  %  ? 

Process.— The  interest  of  $200  for  1  year,  at  6%,      $    $ 
is  $12.     Therefore  if  the  interest  is  $30,  the  time     1J)d(J 
must  be  as  many  years  as  $12  is  contained  times  in  Ans.  2 5  years. 
$30,  or  24  years. 

RULE. — Divide  the  given  interest  by  the  interest  of  the  prin- 
cipal for  1  year. 

Ex.  4.  In  what  time  will  $320  yield  $72.80,  at  7  per  cent.? 

PEINCIPAL. 

Art.  154. —  To  find  the  principal  when  the  interest  or 
amount,  time,  and  rate  are  given. 

Ex.  5.  What  principal,  at  6  %,  will  yield  $30  interest,  in  2 
years  6  mos.  ? 

Process. — $1  at  6  %  will  yield  in  2  years  6  months  $.15      .15^30  00 
interest.      Therefore  $30  interest  will  require  as  many 
dollars  as  $.15  is  contained  times  in  $30,  or  $200. 


COMPOUND   INTEREST.  209 

RULE. — Divide  the  given  interest  or  amount  by  the  interest 
or  amount  of  $l,/br  the  given  time. 

Ex.  6.  What  principal  at  7%  will  yield  $72.80  interest  in 
3  years  3  months  ? 

EXAMPLES. 

7.  At  what  rate  will  $500  yield  $34  interest  in  1  year  1 
month  18  days  ? 

8.  At  what  rate  will  $300  amount  to  $366  in  3  yrs.  8  mo.  ? 

9.  In  what  time  will  $560  yield  $106.40  at  8  per  cent.  ? 

10.  What  principal  will  yield  $192  interest  in  4  yrs.  3  mos. 
6  days,  at  6  per  cent.  ? 

11.  At  what  rate  will  $1200  yield  $3  interest  in  15  days  ? 

12.  In  what  time  will  $360  yield  $10.50  interest  at  5  %  ? 

13.  What  principal  will  yield  $37.50  interest  in  4  years  2 
months,  at  6  %  ? 

14.  In  what  time  will  $360  amount  to  $360.66,  at  6  %  ? 

15.  At  what  rate  will  $350  yield  $101.50  interest  in  7  years 
3  months  ? 

16.  What  principal  will  yield  $9  interest  in  1  year  2  mos. 
12  days,  at6%? 

17.  At  what  rate  will  $500  yield  $62.50  interest  in  2  years  1 
month  ? 

18.  In  what  time  will  $65  yield  $2.60  interest  at  6  %  ? 

19.  What  principal  will  amount  to  $1245  in  3  years  and  6 
months,  at  7  %? 

20.  At  what  rate  will  $1000  amount  to  $1150,  in  2  years  and 
6  months  ? 


COMPOUND  INTEREST. 

Art.  155. — Compound  Interest  is  interest  on  the  prin- 
cipal and  interest  already  due. 

EXAMPLE  1. — What  is  the  compound  interest  of  $500  for  3 
years  8  months,  at  6  %  ? 


210  COMPOUND  INTEREST. 


$500 
.06 


30.00  Interest  1st  year. 
500 

530        Amount  "     " 
.06 


31.80        Interest  2d    " 
530 

561.80        Amount  "     " 


33.7080        Interest  3d    « 
561.80 


595.508          Amount 
.03 


17.86524          Interest  6  months. 
595.508 


613.37324          Amount 
500 


$113.37324  Compound  int'st  for  3  y'rs  6  mos. 

RULE. — Find  the  interest  for  a  year  or  the  time  till  it  is  due, 
and  add  it  to  the  principal  for  a  new  principal ;  on  which 
find  the  interest  as  before.  Proceed  thus  till  the  last  interest 
is  due,  and  from  the  amount  subtract  the  first  principal. 

EXAMPLES. 

What  is  the  compound  interest  of — 
(2.)  $200  for  3  years,  at  7%  ? 
(3.)  $300  for  4  years,  at  6  %  ? 
(4)  $400  for  5  years,  at  5  %  ? 

(5.)  $200  for  2  years,  at  Q%  (payable  semi-ammally  ?) 
(6.)  $100  for  1  year,  at  6%  (payable  quarterly  ?) 

7.  What  is  the  amount  of  700  for  3  years,  9  months,  and  24 
days,  at  7  %,  compound  interest  ? 

8.  What  is  the  amount  of  $740,  at  6  %,  compound  interest, 
(semi-annually,)  from  Dec.  20,  1866,  to  Nov.  2,  1869  ? 

9.  What  is  the  compound  interest  of  $1000  for  2  years,  8 
months,  15  days,  at  6  %? 

10.  What  is  the  amount  of  $500  for  2  years  at  8  %,  com- 
pound interest,  payable  quarterly  ? 


DISCOUNT. 


211 


TABLE, 

Shoioing  the  amount  of  $1,  or  £1,  at  3,  4,  5,  6,  and  7  per  cent  com- 
pound interest,  for  any  nwmber  of  years  from  \  to  10. 


Years 

3  per  cent. 

4  per  cent. 

5  per  cent. 

6  per  cent. 

7  per  cent. 

1 

1.030,000 

1.040,000 

1.050,000 

1.060,000 

1.07,000 

2 

1.060,900 

1.081,600 

1.102,500 

1.123,600 

1.14,4:;0 

3 

.092,727 

1.124,861 

1.157,625 

1.191,016 

1.22,504 

4 

.125,509 

1.169,859 

1.215,506 

1.262,477 

1.31,079 

5 

.159,274 

1.216,653 

1.276,282 

1.338,226 

1.40,255 

6 

1.194,052 

1.265,319 

1.340,090 

1.418,519 

1.50,073 

7 

.229,874 

1.315,932 

1.407,100 

1.503,630 

1.60,578 

8 

1.266,770 

1.368,569 

1.477,455 

1.593,848 

1.71,818 

9 

1.304,773 

1.423,312 

1.551,328 

1.689,479 

1.83,845 

10 

1.343,916 

1.480,244 

1.628,895 

1.790,848 

1.96,715 

Multiply  the  amount  of  $1,  by  the  given  number  of  dollars. 

Find  the  answers  to  the  above  examples  by  the  table. 

The  semi-annual  interest  of  $1  is  the  same  as  the  annual  interest  at 
the  rate  per  cent. 


DISCOUNT. 

Art.  156. — Discount  is  a  certain  percentage  deducted 
for  the  payment  of  money  a  specified  time  before  it  is 
due. 

The  present  worth  of  a  sum,  or  debt,  payable  at  some 
future  time  without  interest,  is  the  sum  which  put  at  in- 
terest till  it  becomes  due,  will  amount  to  the  given  sum. 


EXAMPLE  1.— What  is  the  present  worth  of  $500,  payable  in 
one  year  at  6  per  cent.  ? 


212  DISCOUNT. 

1.06)500(471.698 
424 

760 
24-2 

Process.—  At  6  %  the  present  worth  of  $1.06 
due  in  1  year,  is  $1.00.  Therefore  the  present 
worth  of  $500  is  as  many  dollars  as  it  contains 
times  $1.06,  or  $471.698+.  $500-471.698=  740 

$28.302  discount.  636 

1040 
954 

~860 
848 

RULE.  —  Divide  the  given  sum  by  the  amount  of  $1  for  the 
given  time,  at  the  given  rate  ;  the  quotient  will  be  the  present 
worth.  (Percentage,  Case  IV.) 

To  find  the  discount,  subtract  the  present  worth  from  the 
given  sum  or  debt. 

EXAMPLES. 

What  is  the 

2.  Present  worth  of  $130,  due  in  5  years,  at  6  %? 

3.  Discount  of  $115,  due  in  2  years  6  months,  at  6  %1 

4.  Present  worth  of  $334,  due  in  1  year  1  month  18  days, 
at  6^? 

5.  Discount  of  $666.40,  due  in  2  years  4  months  15  days, 


6.  Present  worth  of  $942,  due  in  4  years  3  months  6  days, 
at  6^? 

7.  Present  worth  of  $534.04,  chie  in  3  years  5  months  18 
days,  at  5  %? 

8.  Discount  of  $366,  due  in  3  years  8  months,  at  6  %? 

9.  Present  worth  of  273.75,  due  in  1  year  7  months,  at  6%? 

10.  What  was  the  discount  of  $263.04,  due  April  27,  1859, 
but  paid  Feb.  15,  1858,  at  8  %  ? 

11.  What  is  the  value  May  10,  1863,  of  a  debt  of  $200,  due 
Aug.  28,  1865,  at  7  %? 

12.  How  much  must  be  paid  July  3,  for  a  debt  of  $142.45, 
due  Nov.  27,  at  9  %? 


BANK   DISCOUNT.  218 

13.  How  much  should  be  deducted  from  a  debt  of  $170.50, 
due  April  19,  1869,  if  paid  Jan.  9,  1867,  at  6  %* 

14.  What  is  the  discount,  Feb.  5, 1862,  on  a  note  of  $407.088, 
payable  Aug.  20,  1864,  at  7  %? 

15.  How  much  should  be  paid,  May  18,  1867,  on  a  note  of 
$5783.09X,  payable  Sept.  25,  1870,  at  8  %? 


BANK  DISCOUNT, 

Art,  157.— Bank  Discount  is  a  certain  percentage  paid 
to  banks,  or  bankers,  for  the  use  of  money  paid  on  notes 
before  they  are  due.  It  is  the  same  as  simple  interest 
paid  in  advance. 

Bank  discount  is  greater  than  true  discount,  because  it 
is  computed  on  the  amount  or  face  of  the  note,  which 
includes  the  interest  with  the  money  lent,  instead  of  only 
the  principal. 

In  computing  bank  discount  three  days  of  grace  are  al- 
lowed, in  addition  to  the  specified  time. 

EXAMPLE  1. — What  is  the  bank  discount  of  a  note  of  $150, 
payable  in  90  days,  at  6  %  ? 

$ 

150. 
.0155 

Process. — The  interest  of  $1  for  93  days  is  $.0155,  ==pr 

and  of  $150,  150X-0155=$2.325.  ?™° 

150 
Ans.  S2.3250 

RULE. — Find  the  interest  for  three  days  more  than  the 
specified  time. 

If  the  note  bears  interest,  find  the  interest  on  the  amount  that  will 
be  due  on  it  at  maturity. 

The  discount  subtracted  from  the  given  sum  gives  the  present 
worth. 


214  EXAMPLES. 

EXAMPLES. 

What  is  the  bank  discount 

2.  On  a  note  of  $300  for  6  months,  at  6  %  ? 

3.  On  a  note  of  $450  for  4  months,  at  5  %  ? 

4.  On  a  note  of  $500  for  3  months,  at  7  %  ? 

5.  On  a  note  of  $750  for  9  months,  at  5 %  %  ? 

6.  On  a  note  of  $1000  for  3  months,  at  7  %  ? 
What  is  the  present  worth 

7.  Of  a  note  of  $120  for  4  months,  at  7  %  ? 

8.  Of  a  note  of  $360  for  30  days,  at  6  %  ? 

9.  Of  a  note  of  $340  for  6  months,  at  8  %  ? 

10.  Of  a  note  of  $480  for  1  month,  at  6  %  ? 

11.  Of  a  note  of  $1950  for  2  months,  at  6  %  ? 

Art.  158. — To  find  for  what  amount  a  note  must  be  given 
that  it  may  be  worth  a  given  sum  when  discounted. 

EXAMPLE  12. — For  what  amount  due  in  60  days  must  a  note 
be  given  that  its  present  worth  may  be  $500,  at  6  %  ? 

Process. — The  bank  discount  of  $1  for  63  days  is  $.0105,  and  the 
present  worth  is  $.9895.     Therefore  $500  is  the  present  worth  of  as 
many  dollars  for  the  same  time  as  $.9895  is  contained  tunes  in  $500. 
$500 -i-  $.9895  =  $505.30. 

RULE. — Divide  the  required  sum  by  the  present  worth  of 

II 

13.  For  what  amount  must  a  note  payable  in  90  days  be 
given  that  its  present  worth  may  be  $300,  at  6  %  discount  ? 

14.  For  what  amount  must  a  note  payable  in  10  months  be 
given  that  its  present  worth  may  be  $500,  at  7  %  discount  ? 

15.  For  what  amount  must  a  note  payable  July  1  be  given 
that  it  may  be  worth  Jan.  1,  the  same  year,  $730,  at  6  %  dis.  ? 

16.  A  man  wishes  to  procure  from  a  bank  $1000.     At  5  % 
discount  what  will  be  the  amount  for  which  he  must  give  his 
note,  payable  in  six  months  ? 


Promiscuous  Examples  in  True  and  Bank  Discount. 

If  bank  discount  is  not  specified,  discount  means  true  discount. 
1.  What  is  the  discount  of  $100  for  six  months  at  6  %  ? 


TAXES.  215 

2.  What  is  the  bank  discount  of  the  same  ? 

3.  What  is  the  present  worth  of  the  same  at  true  discount  ? 
bank  discount  ? 

4.  A  debt  of  $300  is  due  Oct.  1 ;  if  paid  June  1,  the  same 
year,  how  much  should  be  paid  ? 

5.  Bought  $100  worth  of  goods  on  G  months'  credit ;  how 
much  should  be  deducted  for  cash,  at  7  %  discount  ? 

6.  A  note  of  $3000,  payable  in  60  clays,  was  discounted  at  a 
bank  at  6  %  ;  how  much  was  received  for  it  ? 

7.  A  speculator  wished  to  procure  from  a  bank  $10,000  for 
4  mos.     For  what  amount  should  he  have  given  a  note  for  it 
at  1%  discount? 

8.  A  merchant  bought  goods  amounting  to  $3000  on  6  mos. 
credit,  but  was  allowed  5  %  of  the  amount  for  cash ;  money 
being  worth  7  %,  how  much  did  he  gain  by  paying  cash  ? 

9.  A  merchant  bought  75  barrels  of  flour  for  $500  and  sold 
it  for  $640,  receiving  for  it  a  note  payable  in  8  months,  which 
he  had  discounted  at  6  %,  bank  discount ;  how  much  did  he 
gain? 

10.  A  drover  wishes  to  procure  from  a  bank  $2000  for  2  mos. 
15  days  ;  for  what  amount  must  he  give  his  note,  at  7  %  dis.  ? 


TAXES. 

Artt  159. — A  Tax  is  money  required  by  law  to  be  paid 
for  the  support  of  the  government  and  its  institutions,  or 
public  improvements. 

A  poll  tax  is  a  certain  sum  on  male  citizens,  called  polls. 

An  income  tax  is  a  certain  percentage  on  incomes. 

Taxable  property  is  either  Personal  Property  or  Real 
Estate. 

Real  Estate  is  that  which  is  not  movable;  as  land, 
houses,  &c. 

Personal  Property  is  such  as  is  movable,  as  money, 
notes,  furniture,  &c. 


216 


TAXES. 


Taxes  are  commonly  a  certain  percentage  on  property, 
of  all  which  an  inventory,  or  list  of  articles,  is  first  made. 

EXAMPLE  1.  —  In  a  certain  town  $5150  is  to  be  raised  by  tax. 
The  number  of  polls  is  300,  each  taxed  50  cents.  The  real 
estate  is  valued  at  $800,000,  personal  property  $200,  000.  What 
is  the  rate  per  cent,  on  $1,  and  what  is  the  amount  of  A's 
tax,  who  pays  for  5  pells,  and  whose  real  estate  is  valued  at 
$4000  and  personal  property  $1500  ? 

Process.  -300  polls  X  50  cents  =  $150  poll  tax  ;  $5050  —  $150  = 
$5000  property  tax  ;  $5000  -f-  1000000  (800000  -f  200000)  =.005  mills 
the  per  cent,  or  tax  on  $1.  (Percentage,  Case  II.)  A's  property, 
$5500  (4000  +  1500)  X  -005  =  $27.50  A's  property  tax,  to  which  add 
his  poll  tax  (5  polls  X  50  cents)  $2.50.  Ans.  $30.00. 

RULE.  —  Subtract  the  poll  tax  from  the  whole  tax  ;  find  the 
per  cent,  of  the  remainder  on  all  the  property,  and  then  each 
man's  property  tax,  to  which  add  his  poll  tax. 

After  finding  the  tax  on  one  dollar,  it  will  be  convenient  in  prac- 
tice to  make  a  tax  table,  as  follows  : 


Tax  on 


|1.=$.005 
2.=  .01 
3.=  .015 
4.=  .02 
5.=  .025 
6.=  .03 
7.=  .035 


TABLE. 

Tax  on  $8  .=$.04 
9.=  .045 
10.=  .05 
20.=  .10 
30.=  .15 
40.=  .20 
50.=  .25 


Tax  on  $60.  =$.30 

70.=  .35 

80.=  .40 

90.=  .45 

100.=  .50 

1000.  =5. 00 

&c. 


By  this  table  A's  property  tax  in  the  above  example  is  on 
$5000        $25.00 
500  2.50 

$27.50  property  tax. 

Ex.  2.  The  tax  of  a  certain  town  is  $4500.  The  number  of 
polls  is  500,  each  taxed  $1.  The  real  estate  is  valued  at 
$600,000,  and  personal  property  $200,000.  What  per  cent,  is 
the  tax,  and  what  is  B's  tax,  whose  property  is  valued  at 
$3500,  and  who  pays  for  two  polls  ? 

When  district  schools  are  supported  by  families  in  proportion  to 


DUTIES.  217 

the  attendance,  divide  the  whole  expense  by  the  whole  number  of 
days'  attendance,  and  multiply  the  quotient  by  the  days'  attendance 
from  each  family. 


DUTIES. 

Art.  160. — Duties  are  taxes  on  goods  imported  or 
exported. 

A  Port  Of  Entry  is  a  place  where  duties  are  collected. 

A  Custom-House  is  an  office  where  duties  are  collected. 

Ad  valorem  duty  is  a  certain  percentage  on  the  cost 
of  the  goods. 

Specific  duty  is  a  certain  price  per  weight  or  measure. 

An  invoice  is  a  list  of  goods,  with  the  prices. 

Tare  is  an  allowance  for  weight  of  boxes,  casks,  &c., 
containing  the  goods. 

Draft  is  an  allowance  of  weight  for  waste. 

Leakage  is  an  allowance  for  the  waste  of  liquors  in 
casks  or  barrels  ;  breakage,  for  the  same  in  bottles. 

Gross  weight  is  the  weight  of  goods,  including  the 
packages. 

Net  weight  is  the  weight  after  deducting  the  tare,  &c. 

In  all  these  allowances  reject  fractions  less  than  |,  and  add  1  for  3 
or  more. 

EXAMPLE  1. — What  is  the  duty,  at  30  per  cent.,  on  100  gals, 
of  oil,  invoiced  at  75  cts.  a  gallon,  allowing  2  %  for  leakage  ? 

Process.— 100  X  •  02  =      2.     gallons  leakage. 
100—2  =    98  «      net. 

98  X  -75  cts.  =  $73.50  net  value. 
$73. 50  X  •  30         =  $22. 05  duty. 

Ex.  2.  What  is  the  duty  on  10  bbls.  of  sugar,  each  weighing 
215  Ibs.  gross,  at  2  cts.  a  pound — draft  2  Ibs.  each ;  tare  12  >£ 
per  cent.  ? 


218  EXCHANGE. 

Process.— 215  —  2  =      213  pounds. 

213  X  10  =    2130 

2130  X  -12i=      266  Ibs.  tare. 

2130  —  266  =    1864  Ibs.  net. 

1864  X  2  cts-  =$37.28  duty. 

KULE. — Deduct  all  allowances,  then  to  find  the  ad  valorem 
duty  multiply  the  cost  by  the  given  rate  per  cent.;  to  find  the 
specific  duty  multiply  the  net  weight  or  quantity  by  the  duty 
on  ONE  of  the  same. 

Ex.  3.  What  is  the  duty,  at  18  per  cent.,  on  200  bags  of 
coffee,  each  weighing  150  Ibs. ,  invoiced  at  12)^  cts.  a  pound — 
draft  2  Ibs.  each  ;  tare  3  per  cent.  ? 

Ex.  4.  What  is  the  duty,  at  33  >£  per  cent.,  on  36  pieces  of 
silk,  each  containing  45  yards,  invoiced  at  $2  a  yard  ? 

Ex.  5.  What  is  the  duty,  at  10  cts.  a  pound,  on  34  chests  of 
tea,  each  weighing  118  Ibs. — draft  on  each  1  Ib. ;  tare  8  per  ct.  ? 

Ex.  6.  What  is  the  duty,  at  40  per  cent.,  on  500  yards  of 
satin,  at  11.62)^  ? 

Ex.  7.  What  is  the  duty,  at  25  cts.  a  gallon,  on  18  casks  of 
wine,  each  containing  68  gals.,  allowing  2  per  cent,  leakage  ? 

Ex.  8.  What  is  the  duty,  at  12%  cts.  a  pound,  on  12  boxes 
of  tobacco,  each  weighing  130  Ibs. ;  draft  1  pound  on  each  box  ; 
tare  6  per  cent.  ? 

Ex.  9.  What  is  the  duty,  at  30  per  ct. ,  on  9  cases  cf  broad- 
cloth, each  case  containing  20  pieces,  and  each  piece  36  yards, 
at  $4  a  yard  ? 


EXCHANGE. 

Art.  161. — Exchange  is  a  means  employed  by  persons 
in  one  place  of  making  payments  in  another,  such  as 
drafts,  or  bills  of  Exchange.  Exchange  in  the  same 
country  is  Domestic;  from  one  country  to  another, 
Foreign. 


FORMS.  219 

FORM  OF  A  DRAFT. 

^  (^/Lav-en,    C/ct.    /<5, 

&      '__   /    y  (/ 

vaute  tecewea.  and  cnaiae 

/ 

4a?ne  to  tne  account 


/5OO 


E.  Clark  is  the  Drawer. 

S.  Staples  is  the  Drawee. 

Ezra  Jones  is  the  Payee. 

S.  Staples  accepts,  or  promises  to  pay  the  above,  by 
writing  "  Accepted  "  oyer  his  name,  on  the  back. 

If  Ezra  Jones  indorses  it,  by  writing  his  name  on  the 
back,  any  person  who  has  it  is  entitled  to  the  amount. 

Drafts,  or  Bills  of  Exchange,  like  Stocks,  may  be  at 
par,  at  premium,  or  discount,  and  their  value  found  in 
the  same  way.  If  they  are  not  payable  at  sight  they  are 
subject  to  bank  discount,  the  same  as  notes. 

EXAMPLES   OF   DOMESTIC   EXCHANGE. 

1.  What  is  -the  cost  of  the  following  draft,  at  a  premium  of 


.,  Q^aa.  SO,  S&&7. 

' 


cet/i/ed,   and  cnabae  to  tne  account  o/ 
/  / 


220  EXCHANGE  ON  ENGLAND. 


Process.— $1  costs  $1.0025.    Therefore  $300  cost  $1. 0025  X  300  =s= 
$300.75,  Ans. 

2.    What  is  the  cost  of  the  following  draft,  at  2)^  %  dis- 
count ? 


a.  so, 

to 
and      t 

<7 

/k  vauie  kecewea,  and  cnaiae  to  6ne  account 


Process.—  Present  worth  of  $1,  by  bank  discount=$.9936;  $.9936— 
.025—  $.9686  the  cost  $1  of  the  draft,  $.9686X.480=$464.928,  Ans. 

3.  A  merchant  in  St.  Louis  wishes  to  remit  a  draft  for  $1000 
to  New  York  ;  what  will  it  cost,  exchange  being  at  2}^  per 
cent,  premium  ? 

4.  A  merchant  in  New  Orleans  wishes  to  remit  to  Philadel- 
phia a  draft  for  $1500,  at  thirty  days'  sight  ;  what  will  it  cost, 
exchange  being  at  4  per  cent,  premium  at  sight. 

Drafts  are  usually  drawn  at  30,  60,  or  90  days'  sight,  and  at  a  cer- 
tain per  cent,  premium  or  discount,  including  allowance  for  the 
time. 

EXCHANGE  ON  ENGLAND. 

Art.  162.  —  Exchange  on  England  is  always  at  a  pre- 
mium in  this  country,  because  in  making  it,  a  pound 
sterling  is  valued  at  $444f,  instead  of  its  true  value, 

$4.84 


PEOMISCUOUS  EXAMPLES.  221 

Ex.  1.  —  BILL  OF  EXCHANGE  ON  ENGLAND. 


and   wwia    o/  Aame  aafa  and 
S 

nfoatd,/     /iau   to 
'          /     / 


wn 


What  is  the  cost  of  the  above  bill,  when  exchange  is 
premium  ? 

Process.—  £1=S4|  (old  value)  £300X^=$1333.33i  $1  at 
premium=$1.105.     $1333.  33£><1-  105^$M73.33i,  Ans. 


Art.  163.  —  To  Find  {he  Cost  of  a  Bill  of  Exchange  on 
England. 

EULE.  —  Reduce  the  pounds  and  decimal  of  a  pound,  at  the 
old  value,  (  $4fJ  to  dollars,  &c.;  which  multiply  by  the  cost 


Ex.  2.  A  gentleman,  about  to  visit  England,  wishes  to  buy 
a  bill  exchange  for  $2000  ;  what  will  be  the  amount  of  it  at 
9}£  premium? 

Process.—  $1.095  =  cost  of  $1. 

$1.095  X  44  =  $4.86|  the  cost  of  £1. 

$2000  ~  4.  86|  =  £410.959  +  =  £410  19s.  2d.  + 

Art.  164,  —  To  find  the  amount  of  a  Bill  on  England 
which  can  be  bought  with  U.  S.  Money. 

RULE  —  Divide  the  given  number  of  dollars  by  the  cost  of 
£1. 


222  PARTNERSHIP. 

EXAMPLES  IN  EXCHANGE  ON  ENGLAND. 

3.  What  will  a  bill  of  exchange  on  London,  amounting  to 
£500,  cost  in  New  York,  exchange  being  9%  per  cent.  prem.  ? 

4.  A  gentleman  wishes  to  buy  a  bill  of  exchange  on  Liver- 
pool with  $2500 ;   what  will  be  its  amount  at  12  %  %  pre- 
mium ? 

5.  An  importer  wishes  to  pay  for  goods  ordered  from  Man- 
chester, England,  amounting  to  £1000;  what  will  a  bill  of 
exchange  cost  at  9^  per  cent,  premium  ? 

6.  An  importer  wishes  to  remit  to  Leeds,  England,  $3000  ; 
what  will  be  the  amount  of  a  bill  of  exchange,  at  10  per  cent, 
premium.  ? 

Exchanges  on  other  foreign  countries  are  made  by  reduction  of 
currencies. 


Promiscuous  Examples  in  Exchange. 

1.  What  is  the  cost  of  a  draft  on  Memphis  for  $3600,  at  1% 
per  cent,  premium  ? 

2.  What  is  the  amount  of  a  draft  which  costs  $1012.50,  at 
1£  per  cent,  premium  ? 

3.  What  will  a  draft  on  New  York  for  $600  cost,  at  2  per 
cent,  discount? 

4.  What  will  be  the  cost  of  a  bill  of  exchange  on  Liverpool, 
England,  of  £150  10s.,  at  9£  per  cent,  premium  ? 

5.  What  will  be  the  amount  of  a  bill  of  exchange  on  Lon- 
don, bought  in  Boston  for  $3000,  at  9^  per  cent,  premium  ? 

6.  What  will  be  the  cost  of  a  draft  for  $750,  on  Hartford,  at 
^  per  cent,  discount  ? 

7.  What  is  the  amount  of  a  draft  bought  for  $1250,  at  a 
premium  of  2^  per  cent.  ? 


EXAMPLES.  223 


PARTNERSHIP. 

Art.  165. — Partnership  is  a  company  of  two  or  more 
persons  in  the  same  business.  They  are  called  a  firm, 
or  house,  and  each  member  a  partner. 

The  capital,  or  stock,  is  the  money  or  property  em- 
ployed in  their  business,  of  which  the  profit  or  loss  is  a 
certain  part  or  percentage. 

In  Bankruptcy  the  creditors  are  the  same  as  partners 
in  business,  and  the  property  of  the  bankrupt  as  profit 
or  loss. 

CASE  I. 

Art.  166. — To  find  each  partner's  share  of  the  gain  or  loss 
when  their  capital  has  been  used  the  same  length  of  time. 

EXAMPLE. — Peck,  Staples  &  Clark  are  partners  in  business. 
P.'s  capital  is  $4800  ;  S.'s  $2400,  and  C.'s  $1800.  The  whole 
gain  is  $3000  ;  what  is  each  one's  share  ? 

Process.-P.'s  capital,  $4800  \  The  whole  gain  is  IW=  i.  or 
S  's      "        $2400  333  Per  cent-  of  the  wllole  caP~ 

r '«      "        SfftMl  -I       ital-     Therefore  each  partner's 
f±^  share  is  i,  or  33^%  of  his  capi- 

Whole  capital,    &9000  tal      Therefore, 

$4800  X  i  or  -33^=  $1600  P.'s  gain,  ) 

$2400'  "  $800  S.'s     "      [  Ans. 

$1800  "  S  600  C.'s     "      ) 

RULE. — Take  such  a  part  of  each  partner's  capital  as  the 
whole  gain  or  loss  is  of  the  whole  capital";  or 

Multiply  each  partner's  capital  by  the  gain  or  loss  per  cent. 

EXAMPLES. 

2.  Messrs.  Staples  &  Clark  are  in  partnership  ;  S.  's  capital 
is  $3000,  C.'s  $2000;  they  have  gained  $1500;  what  is  each 
one's  share  ? 

3.  Messrs.    Howland  &  Son  shipped  a  cargo   of    goods, 
amounting  to  $20,000,    %  of  which  belonged  to  H.      The 
profits  were  $4000  ;  what  was  each  one's  share  ? 

4.  The  capital  of  a  Eailroad  Company  is  $2,000,000,  the 


224  PROMISCUOUS  EXAMPLES. 

annual  earnings  are  $400,000,  the  expenses  $240,000.     I  own 
25  shares  ;  what  is  my  annual  dividend  ? 

5.  A  factory  owned  by  three  men  was  damaged  by  fire  to 
the  amount  of  $3000  more  than  the  insurance.     S.  owned  £, 
E.  f ,  and  W.  the  remainder ;  what  was  each  man's  share  of 
the  loss  ? 

6.  A  merchant  having  failed  in  business,  owed  J.  Strong 
$825,  T.  Williams  $700,  S.  Vernon  $1175.     He  can  pay  the 
three  only  $900  ;  what  is  each  one's  share  ? 

CASE  H. 

Art.  167. — To  find  each  partner's  share  of  the  gain  or 
loss,  when  their  capital  has  been  used  unequal  portions  of 
time. 

EXAMPLE  7. — Mead,  Rogers  &  Smith  have  furnished  capital 
as  f oUows :  Mead,  $5000  for  2  months ;  Eogers,  $4000  for  4 
months,  and  Smith  $3000  for  3  months.  They  have  gained 
$3500 ;  what  is  each  one's  share  ? 

Mead's  capital,  $5000  X  2  =  S 10000  for  1  month. 
Kogers'      "        $4000  X  ±  =  $16000 
Smith's      "        $3000X3=$  9000 
The  whole        "  =  $35000          " 

The  whole  gain  is  ^ftftfty  =  -,L0-  of  the  sum  of  the  products  of  each 
one's  capital  multiplied  by  the  time  it  was  used  ;  therefore  each 
partner's  share  is  -fa,  or  10  per  cent,  of  his  capital,  multiplied  by  its 
time. 

Mead's  share  is  A  of  $10000  =  $1000  ) 
Bogers'  share  is  -^  of  $16000=  $1600  }  Ans. 
Smith's  share  is  ^  of  $  9000  =  $  900  ) 

KULE. — Multiply  each  partner's  capital  by  the  time  it  was 
used,  and  treating  the  product  as  his  capital,  proceed  as  in 
Case  I. 

EXAMPLES. 

8.  Messrs.  Hoyt  &  Lane  formed  a  partnership  as  drovers ; 
H.  furnished  $3200  for  2  months,  and  L.  $2000  for  4  months. 
They  gained  $1350  ;  what  was  each  one's  share  ? 

9.  Two  carpenters,  Adams  &  Nelson,  contracted  to  build  a 


PROMISCUOUS  EXAMPLES.  225 

house  for  $600  ;  A.  furnished  9  men  for  100  days,  and  Nelson 
12  men  for  75  days  ;  what  was  each  one's  share  ? 

10.  Three  men  performed  a  piece  of  work  in  28  days  alto- 
gether, for  which  they  were  paid  $42.     A.  received  $15,  B. 
$12,  and  C.  the  remainder ;  how  many  days  did  each  work  ? 

11.  Messrs.  Todd  &  Howe  engaged  in  business  with  a  capi- 
tal of  $3600.     T.'s  capital  was  in  the  business  4  months,  and 
his  share  of  the  profits  was  $160  ;  B.  's  capital  was  in  the  busi- 
ness 6  months,  and  he  received  as  his  share  $192  ;  how  much 
capital  did  each  furnish  ? 


Promiscuous  Examples  in  Partnership. 

1.  Messrs.  Tappan,  Edwards  &  Kimballwere  partners.     T. 
furnished  $800  capital,  E.   $700,  K.  $1300  ;  they  lost  $560 ; 
what  was  the  loss  of  each  ? 

2.  A  man  bequeathed  $8400  to  his  three  sons,  in  proportion 
to  their  ages,  10,  14,  and  18  years ;  how  much  did  each  re- 
ceive ? 

3.  Messrs.   Hawley,   King  &  Stebbins  were  partners.     H. 
furnished  $10,000  capital  for  15  months,    K.   $12,000  for  a 
year,  and  S.  $15,000  for  9  months.     They  gained  $9652.50  ; 
what  was  each  man's  share  ? 

4.  Messrs.  Benedict  &  Coe  were  partners  three  years.     B. 
furnished  $6150  capital,  C.  $8100  ;  B.'s  share  of  the  gain  was 
$1250  ;  what  was  the  whole  gain  ? 

5.  C.  Jenkins  commenced  business  with  a  capital  of  $2500  ; 
after  6  months  he  took  into  partnership  A.  Heed,  with  $3503 
capital ;  3  years  afterwards  their  joint  capital  was  doubled ; 
how  much  had  each  gained  ? 

6.  I  agreed  to  pay  three  men  $72  for  delivering  to  families, 
at  different  distances,  36  tons  of  coal.     A  drew  9  tons  6  miles. 
B   12  tons  4  miles,  C  thf  jggmainder,  2  miles  ;  how  much  is 
due  to  each  ? 

7.  Mesdz&Scott  &  Taylor  were  partners  three  years ;  S.  fur- 
nished %  as  much  capital  as  T. ;  they  gained  $5000  ;  what  was 
each  one's  share  ? 


226  PERCENTAGE. 

8.  Three  drovers  hired  a  pasture  for  $36.     A  had  24  head  of 
cattle,  and  paid  $12  ;  B  paid  $16,  and  C  the  remainder,  as 
their  portions  ;  how  many  cattle  did  B  and  C  each  have  in  the 
pasture  ? 

9.  Three  drovers  had  500  sheep  each,  for  which  they  hired 
a  pasture  and  paid  $56,  each  agreeing  to  pay  in  proportion  to 
the  number  of  weeks  his  sheep  were  in  the  pasture.     A  paid 
$14  for  2  weeks,  B  paid  $17^,  and  C  $24^  ;  how  long  were 
B  and  C's  sheep  in  the  pasture  ? 

10.  A  bankrupt  owes  $18,000,  and  his  property  is  worth 
$3600  ;  how  much  will  a  creditor  receive,  whom  he  owes  $1500? 

11.  Messrs.  Paige,  Cassidy  &  Warren  are  partners.     P. 's 
capital,  $20,000,  has  been  in  the  business  3  years ;  C.'s  capi- 
tal, $15,000,  2  years  6  months ;  W.'s  capital,  $12,000,  1  year  9 
months  ;    they  have  gained  $2962)^  ;    what  is  each  man's 
share  ? 


Art,  168.— Promiscuous  Examples  in  the  various 
applications  of  Percentage. 

EXEECISE   I. 

1.  At  2j^  %  commission,  how  much  will  an  agent  receive 
for  selling  goods  amounting  to  $920  ? 

2.  A  commission  merchant  has  received  $624  for  the  pur- 
chase of  goods  after  deducting  4  %  commission  ;  what  amount 
must  he  expend  ? 

3.  At  3  %  commission,  what  amount  of  goods  must  be  sold 
in  a  year  to  realize  an  income  of  $2400  ? 

4.  What  is  the  premium  on  an  insurance  policy  of  $4500,  at 

iy*%t 

5.  For  what  must  a  house  valued  at  $5000,  be  insured  at 
2  %,  to  cover  the  cost  of  insurance  ? 

6.  What  is  the  value  of  15  shares  of  bank  stock,  at  3  %  be- 
low par  ? 

7.  What  will  a  broker  receive  for  exchanging  $578  in  bank 
notes,  at  1)£  %  discount  ? 


PROMISCUOUS   EXAMPLES.  227 

8.  What  will  a  broker  give  for  $720  in  bank-notes  at  %  % 
discount  ? 

9.  What  is  the  interest  of  $1500  for  4  years  3  months  6  days, 


10.  What  is  the   amount  of  $360  from  June  14,  1863,  to 
Sept.  28,  1865,  at  7  %? 

11.  A  note  of  $300  was  dated  Jan.  1,  1863.     Interest  6  %. 
Indorsed  July  1,  1863,  $109.     Jan.  1,  1864,  $100.     What  was 
due  July  1,  1864  ? 

12.  A  man  has  paid  in  3  years,  $341.75  interest,  at  5%; 
what  was  the  principal  ? 

13.  A  man  pays  $800  rent  for  a  house  valued  at  $10,000  ; 
what  per  cent,  interest  does  he  pay  ? 

14.  What  is  the  compound  interest  of  $600  for  2  years,  6 
months,  at  6  %  ? 

15.  What  is  the  present  worth  of  $399.60,  due  in  1  year  10 
months,  at  6  %  discount  ? 

16.  What  is  the  bank  discount  on  $750,  payable  in  one 
month  ? 

17.  What  is  the  present  worth  of  a  note  for  $360,  payable  in 
1  month,  at  6  %  bank  discount  ? 

18.  For  what  sum  must  a  note  at  6  %  for  90  days,  be  given 
at  a  bank  to  obtain  $393.80  ? 

19.  A  merchant  sold  goods  amounting  to  $300,  and  gained 
20  %;  what  did  he  gain  ? 

20.  If  silk  cost  $1.80  a  yard,  for  what  must  it  fee  sold  to  gain 
25^? 

21.  A  man  paid  $75  for  a  wagon,  and  sold  it  for  $100  ;  what 
per  cent,  did  he  gain  ? 

22.  A  man  sold  a  wagon  for  $112,  and  gained  40  %,  what 
did  it  cost  ? 

23.  A  town  is  taxed  $6250  on  its  property  valued  at  $1,  200,  000, 
and  there  are  500  polls  taxed  50  cts.  each  ;  what  per  cent  is 
the  tax  on  the  property  ? 

24.  What  is  the  duty,  at  20  %  ad  valorem,  on  80  bales  of 
imported  wool,  each  weighing  400  Ibs.  ,  invoiced  at  25  cts.  per 
pound  ;  tare  5  %1 


228  PERCENTAGE. 

25.  Me  Kae  &  Oakley  shipped  goods  in  partnership.    Me  Kae 
furnished  $5000  ;    Oakley,  $3000  ;  they  gained  $2320  ;    what 
was  each  one's  share  ? 

26.  When  gold  is  130,  what  is  $100  in  currency  worth  ? 

27.  What  will  be  the  cost  of  a  draft  at  sight  on  St.  Louis  for 
$1000,  at  1  %  premium  ? 

28.  What  must  be  paid  for  a  bill  of  exchange  on  Liverpool, 
Eng.,  for  £400,  at  9  %  premium  ? 

EXERCISE  H. 


29.  A  man  having  $8,000  lost  12)£  %  of  it  ;  how  much  had 
he  left  ? 

30.  A  steamboat  is  insured  for  $30,000  at  2)£  %\ 
premium  ? 

31.  What  will  20  shares  of  the  Central  Kailroad  cost  at 
%  advance  ? 

32.  A  bank  has  failed  whose  circulation  is  $75,000,  and  is 
able  to  pay  only  66%  %\  what  amount  can  it  pay  ? 

33.  What  is  the  duty  on  840  bags  of  coffee,  each  weighing 
120  Ibs.  at  3  cts.  a  pound  ;  tare  3  per  cent.  ? 

34.  The  expenses  of  a  school  district  are  for  teacher's  salary 
$600  ;  fuel,  &c.,  $37.     The  whole  attendance  has  been  5600 
days  ;  what  must  a  man  pay  for  80  days'  attendance  ? 

35.  Bought  195  cords  of  wood  at  $4.12)^  a  cord,  and  sold  it 
for  $4.87>£  a  cord  ;  what  was  the  gain  per  cent.? 

36.  A  gentleman  sold  a  carriage  for  $150,  which  was  75  % 
less  than  it  cost  ;  what  did  it  cost  ? 

37.  If  I  pay  $960  for  wheat,  for  what  must  I  sell  it  to  gain 
25  per  cent.  ? 

38.  What  is  the  interest  of  $691.20  from  March  5,  1862,  to 
Sept.  20,  1864,  at  7  per  cent.  ?     , 

39.  How  much  money  at  6  %  interest  will  yield  a  semi-an- 
nual income  of  $650  ? 

40.  What  is  the  interest  of  $350,  at  6  %t  for  1  year  6  mos. 
12  days  ?    Also,  the  discount  and  bank  discount  of  the  same 
for  the  same  time  ? 


PEOMISCUOUS  EXAMPLES.  229 

41.  $87.  25  NEWBUEGH,  N.  Y.,  Aug.  14,  1857. 
Six  months  after  date  I  promise  to  pay  H.  S.  Barnes  Eighty- 

seven  ^^  dollars,  value  received,  with  interest. 

EOBT.  LIVINGSTON. 

Indorsed,  Dec.  8,  1858,  $5.00.     May  20,  1859,  $45.50. 
How  much  was  due  August  26,  1859  ? 

42.  What  is  the  compound  interest  of  $200  for  3  years  at  7 
per  cent.  ? 

43.  What  is  the  value  of  a  draft  on  New  Orleans  for  $1500, 
at  3  %  discount  ? 

44.  When  gold  is  125,  how  much  of  it  is  equal  to  $500  in 
currency  ?  how  much  currency  is  equal  to  $500  in  gold  ? 

45.  For  what  amount  will  $2427.77|  purchase  a  bill  of  ex- 
change on  Liverpool,  at  9^  per  cent.  ? 

46.  Messrs.  Taylor  &  Olmsted  are  partners  in  business.     T. 
furnished  $5000  capital  for  6  months,  O.  $4000  for  5  months. 
They  have  gained  $2500  ;  what  is  each  one's  share  ? 

EXERCISE   TTT. 

47.  A  merchant  having  shipped  flour  amounting  to  $1260, 
paid  2^  %  for  insurance  ;  what  was  the  whole  cost  ? 

48.  An  auctioneer  sold  goods  amounting  to  $956.20,  at  2)£ 
per  cent,  commission  ;  how  much  did  he  receive  ? 

49.  A  merchant  paid  a  broker  $3090  for  a  draft  on  Charles- 
ton, allowing  him  3  %  brokerage  ;  what  was  the  amount  of  the 
draft  ? 

50.  What  is  the  amount  due  on  a  note  of  $391,  dated  Sept. 
7,  1864,  and  payable  Feb.  1,  1865,  at  6  per  cent.  ? 

51.  $1000.  UTICA,  May  1,  1862. 
For  value  received  I  promise  to  pay  H.  Low  &  Co.,  or  order, 

One  Thousand  Dollars,  with  interest  at  7  %. 
Indorsed :— Nov.  1,  1862,  $150.     March  13,  1863,  $200. 
What  was  due  Sept.  19,  1863  ? 

52.  A  man  at  his  death  bequeathed  $8000  to  an  asylum,  to 
be  paid  after  it  amounted  to  $10000,  at  1  %  ;  how  long  was  the 
time? 


230  PERCENTAGE. 

53.  What  is  the  interest  of  £113  10s.  for  1  year  6  months,  at 
6  per  cent.  ? 

54.  What  is  the  compound  interest  of  $250  for  3  years,  at  6 
per  cent.? 

55.  What  is  the  present  worth  of  $500.76  payable  in  8  inos., 
at  6  per  cent,  per  annum  ? 

56.  What  is  the  present  worth  of  the  same  by  bank  dis- 
count ? 

57.  What  is  the  difference  between  the  discount  and  the 
interest  of  $1000  for  1  year,  at  6  per  cent.  ? 

58.  For  what  amount  must  a  note  for  90  days  be  given  at  a 
bank  to  obtain  $600,  at  7  %  ? 

59.  Shipped  5  loads  of  furniture,  worth  $1200,  to  Mobile, 
and  paid  2)^  %  for  insurance  ;  what  was  the  premium  ? 

60.  For  what  amount  must  the  same  furniture  have  been 
insured  to  have  covered  the  whole  loss  if  the  ship  had  sunk  ? 

61.  If  a  man  pays  $90  premium  at  2)^  %  for  insurance,  for 
what  amount  is  he  insured  ? 

62.  A  gentleman  bought  a  harness  for  $60,  and  afterwards 
sold  it  for  12  %  less  ;  how  much  did  he  lose  ? 

63.  Bought  a  firkin  of  butter  for  $26.63  ;  for  how  much 
must  it  be  sold  to  gain  20  %  ? 

64.  A  man  bought  a  farm  at  $85  an  acre,  and  afterwards 
sold  it  for  $100  an  acre ;  what  per  cent,  did  he  gain  ? 

65.  A  gentleman  sold  his  watch  for  $72,  which  was  28  per 
cent,  less  than  it  cost ;  what  did  it  cost  ? 

66.  What  is  the  duty,  at  2  cents  a  pound,  on  10  boxes  of 
sugar,  each  weighing  125  Ibs.,  allowing  6  Ibs.  a  box  for  tare  ? 

67.  What  is  the  duty  on  a  cargo  of  coffee,  invoiced  at  $3560, 
at  30  per  cent.  ? 

68.  What  is  the  amount  of  tax  due  from  J.  Carpenter,  whose 
property  is  assessed  at  $1800,  and  who  pays  for  three  polls,  the 
whole  tax  of  the  town  being  $3600  ;  the  number  of  polls  602, 
paying  75  cents  each,  and  the  amount  of  property  assessed 
is  $250,000  ? 

69.  How  many  shares  of  the  Erie  Eailroad  stock  can  be 
bought  for  $7500,  when  it  is  37)^  %  below  par  ? 


PROMISCUOUS  EXAMPLES.  231 

70.  When  gold  is  129)^,  how  much  of  it  must  be  paid  for 
$750  in  currency  ?    How  much  currency,  also,  is  equal  to 
$750  in  gold  ? 

71.  What  is  the  value  in  Memphis  of  a  draft  on  New  York 
for  $3200,  &i2%%  premium  ? 

72.  What  must  be  paid  for  a  bill  of  exchange  on  England 
for  £300  10s.,  at  9^  per  cent,  premium  ? 

73.  A  bankrupt's  debts  amount  to  $30,000  ;  the  property  in 
his  possession  $18,000  ;  how  much  can  be  paid  a   creditor 
whom  he  owes  $1251.37}^  ? 

74.  Messrs.   Halsey  &  Coe  are  partners  in  business;  H.'s 
capital  is  $5000,  and  has  been  employed  3  years  ;  C.  's  capital 
is  $4500,  but  was  not  paid  in  till  4  months  after  H.  's  ;  they 
have  gained  $5400 ;  what  is  each  one's  share  ? 

EXERCISE  IV. 

75.  The  population  of  a  city  has  increased  250  per  cent, 
in  20  years,  and  now  contains   175,000  ;  what  was  its  popula- 
tion 20  years  ago  ? 

76.  A  lawyer  charges  4  %  commission  for  collecting  a  debt 
of  $625.25  ;  how  much  will  he  receive  ? 

77.  A  country  merchant  has  forwarded  $3681  to  his  agent  in 
the  city  for  the  purchase  of  goods,  after  deducting  2^  %  com- 
mission ;  how  much  will  be  the  commission  ? 

78.  What  is  the  premium  for  insuring  a  mill  valued  at  $4620, 
at  1%  per  cent.? 

79.  For  what  sum  must  a  house,  valued  at  $3800,  be  insured, 
at  2  %  to  cover  the  property  and  premium  ? 

80.  What  is  the  value  of  16  shares  in  a  Manufacturing  Com- 
pany, at  12)£  %  premium  ? 

81.  What  must  be  paid  for  a  draft  for  $1275.25,  at  %  % 
premium  ? 

82.  What  is  the  amount  of  $1475.28  from  March  29,  1866,  to 
July  4,  1867,  at  7  per  cent.  ? 

83.  $200.  NEW  HAVEN,  July  18,  1861. 
Three  months  after  date  I  promise  to  pay  J.  Atwater,  or 

order,  Two  Hundred  Dollars,  with  interest,  at  6  %f  value  re- 
ceived. B.  CLAEK. 


232  PERCENTAGE. 

Indorsed  :—  Jan.  18,  1864,  $20.  July  1,  1867,  $60. 
What  was  due  Nov.  7,  1867  ? 

84.  A  man  wished  to  lea.ve  his  daughter  an  income  of  $600  a 
year  ;  what  principal  would  be  required  at  7  %  interest  ? 

85.  What  is  the  amount,  at  compound  interest,  of  $500  for 
2  years,  at  5  %  ? 

86.  What  is  the  present  worth  of  $800,  due  in  4  years  and  8 
months,  at  6  %  ? 

87.  What  is  the  discount  of  $800,  due  in  10  years,  at  6  %  ? 

88.  What  is  the  bank  discount  of  $800,  due  in  10  years,  at 


89.  For  what  sum  must  a  note,  payable  in  60  days,  be  given 
to  obtain  from  a  bank  $800,  at  6  %  discount  ? 

90.  Bought  a  span  of  horses  for  $400,  and  sold  them  at  25% 
profit  ;  for  what  were  they  sold  ? 

91.  Bought  a  span  of  horses  for  $400,  and  sold  them  for 
$450  ;  what  per  cent,  was  gained  ? 

92.  Sold  a  span  of  horses  for  $400,  which  was  25  %  less  than 
they  cost  ;  what  did  they  cost  ? 

93.  Sold  a  span  of  horses  for  $400,  which  was  25  %  more 
than  they  cost  ;  what  did  they  cost  ? 

94.  A  man  whose  property  was  valued  at  $6240,  and  who 
paid  for  three  polls  at  $1  each,  was  taxed  $34.  20;  what  per 
cent,  tax  did  he  pay  ? 

95.  What  is  the  duty,  at  12  %,  on  20  boxes  of  tobacco,  each 
weighing  250  Ibs.  ,  and  costing  20  cents  a  pound  ;  tare  6^  %  ? 

96.  Three  men  bought  a  farm  of  300  acres,  and  agreed  to 
divide  it  according  to  the  amount  each  one  could  pay.    A  paid 
$12,000,  B  $10,000,  C  $8,000;  how  many  acres  did  each  one 
have  ? 

97.  When  gold  is  135,  what  is  the  value  of  $270  in  currency  ? 
How  much  currency  must  be  given  for  $270  in  gold  ? 

98.  For  what  amount  can  a  bill  of  exchange  on  Liverpool, 
England,  be  purchased  with  $720,  at  8  %  premium  ? 

EXEECISE  v. 

99.  If  a  city,  with  a  population  of  120,000,  should  increase 
10  %  a  year,  what  would  be  its  population  after  10  years  ? 


PROMISCUOUS   EXAMPLES.  233 

100.  An  auctioneer  having  sold  some  goods,  retained  £?..08 
and  paid  the  owner  $3492  ;  what  per  cent,  commission  did  he 
charge  ? 

101.  A  broker  received  $2412  for  the  purchase  of  stocks, 
after  deducting  %  per  cent,  brokerage  ;  how  many  $100  shares 
did  he  buy  ? 

102.  What  is  the  premium  for  the  insurance  of  a  factory, 
valued  at  $5400,  at  2%  per  cent.? 

103.  For  what  must  a  house,  valued  at  $1987.50,  be  insured, 
a^  %  %>  to  cover  both  the  house  and  premium  ? 

104.  How  many  shares  of  bank  stock,  at  30  %  premium,  can 
be  bought  for  $1300  ? 

105.  When  gold  is  130,  how  much  of  it  can  be  bought  for 
$650  in  currency  ?    How  much  currency  is  equal  to  $650  in 
gold? 

106.  What  is  the  interest  of  $720.50  from  Jan.  16,  1865,  to 
July  31,  1865,  at  7  %  ? 

107.  What  is  the  discount,  Jan.  16,  1865,  of  $720.50,  due 
July  31,  1865  ? 

108.  What  is  the  bank  discount,  Jan.  16,  1865,  of  $720.50, 
due  July  31,  1865  ? 

109.  $500.  NORWALK,  July  25,  1865. 
On  demand,  for  value  received,  I  promise  Wm.  Kellogg,  or 

order,  Five  Hundred  Dollars,  with  interest. 

J.  CARTER. 

Indorsed :— Dec.  19,  1866,  $62.     Aug.  4,  1867,  $48. 
What  was  due  Oct.  8,  1867  ? 

110.  What  is  the  compound  interest  of  $400,  for  2  years  8 
months,  at  5  %  ? 

111.  If  cloth  cost  $5.50  a  yard,  for  what  must  it  be  sold  to 
gain  20  per  cent.  ? 

112.  If  cloth  cost  $5.50  a  yard,  and  is  sold  for  $3,  what  per 
cent,  is  gained  ? 

113.  If  cloth  is  sold  for  $6,  and  the  gain  is  25  %,  what  did 
it  cost  ? 

114.  What  is  the  duty  on  cutlery  invoiced  at  $2400,  at  30  %t 

115.  The  tax  levied  on  a  town  is  $3440,  the  number  of  polls 


234  PERCENTAGE. 

is  1000,  at  $1  each ;  the  inventory  of  property  is  $320, 000 ; 
what  is  a  man's  tax  whose  property  amounts  to  $2400,  and 
who  pays  for  2  polls  ? 

116.  What  must  be  given  for  a  bill  of  exchange  on  London 
for  £600  12s.,  at  $%  %  premium  ? 

117.  A  bankrupt  owes  $17,000,  and  is  able  to  pay  only  9000 ; 
how  much  will  a  man  lose  whom  he  owes  $680  ? 

118.  Messrs.  Carter,  Bo  we  &  Foster  are  partners  in  business. 
C.  furnishes  ^  the  capital,  E.  £,  and  F.  the  rest ;  they  have 
gained  $3300  :  what  is  each  one's  share  ? 

EXERCISE  VI. 

119.  A  merchant  having  a  capital  of  $6480,  lost  25  %  of  it ; 
how  much  had  he  left  ? 

120.  A  farmer  having  purchased  some  land,  spent  10  %  of 
the  price  in  improvements,  and  then  found  that  the  whole 
cost  was  $8800  ;  what  did  he  pay  for  the  land  ? 

121.  A  merchant  having  lost  10  %  of  his  capital,  had  $1800 
left ;  how  much  had  he  at  first  ? 

122.  A  merchant  having  $1800  capital,  lost  $300  ;  what  per 
cent,  was  his  loss  ? 

123.  Bought  100  tons  of  iron  for  $3865,  and  sold  it  at  3%  % 
less  than  cost ;  what  was  the  selling  price  ? 

124.  Bought  500  bushels  of  wheat  for  $1300,  and  sold  it  for 
$1560  ;  what  per  cent,  was  the  gain  ? 

125.  Sold  500  bushels  of  corn,  and  gained  $100,  which  was 
20  %  of  the  cost ;  what  was  paid  for  it  ? 

126.  Borrowed,  Jan.  1,  in  New  York,  $1250,  and  returned  it 
with  interest  the  following  Sept.  15  ;  what  was  the  amount  re- 
turned ? 

127.  What  is  the  interest  of  $340  from  July  20,  1860,  to  July 
2,  1861,  at  7  per  cent.? 

128.  What  is  the  interest  of  £760  5s.  6d.,  at  6  %,  for  2  years 
4  months  ? 

129.  What  is  the  amount  of  $850  for  5  years  6  months  3 
days,  at  6  %  ? 


PROMISCUOUS   EXAMPLES.  235 

130.  $500.  HAKTFOKD,  Jan.  16,  1855. 
On  demand,  I  promise  to  pay  A.  Terry,  or  order,  Five  Hun- 
dred Dollars,  for  value  received,  with  interest.      J.  E.  DAY. 

Indorsed  .-—April  1,  1856,  $50.     July  16,  1857,  $400.     Sept. 
1,  1858,  $60. 
What  was  due  Nov.  16,  1858  ? 

131.  A  gentleman  has  a  son  15  years  old,  and  he  wishes  to 
invest  for  him  such  a  sum  as  will  amount  to  $10,000  when  he 
is  of  age  ;  what  sum  must  be  invested,  at  7  per  cent.  ? 

132.  A  young  man  has  received  a  legacy  which  yields  a 
semi-annual  income  of  $750,  at  6  per  cent. ;  what  is  the  amount 
of  the  legacy  ? 

133.  A  man  left  his  son  $6000,  possession  to  be  given  after 
it  amounts  to  $9000,  at  6  per  cent. ;  how  long  must  the  son 
wait  for  it  ? 

134.  A  young  man  has  a  legacy  of  $622.75,  to  be  paid  in  3}£ 
years,  but  he  wishes  to  have  it  immediately  ;  what  is  it  worth 
at  5  %  discount  ? 

135.  What  is  the  present  value  of  a  note  for  $900,  payable 
in  6  months,  at  6  per  cent,  bank  discount  ? 

136.  What  is  the  present,  value  of  a  note  for  $900,  payable 
in  6  months,  with  interest,  at  6  %,  provided  the  discount  at  a 
bank  is  7  per  cent.  ? 

137.  For  what  amount  must  a  note,  payable  in  60  days,  be 
given  to  obtain  from  a  bank  $400,  at  6  %  ? 

138.  An  auctioneer  receives  $112.50  for  selling  goods,  at  2% 
commission  ;  what  was  the  amount  sold  ? 

139.  A  commission  merchant  has  received  $1656  for  the  pur- 
chase of  goods,  after  deducting  3}^  %  commission ;  what  is 
the  amount  of  the  goods  to  be  purchased  ? 

140.  What  is  the  value  of  50  shares  of  the  Illinois  Central 
Eailroad,  at  a  premium  of  9  per  cent.  ? 

141.  How  many  shares  of  stock,  10,%  below  par,  can  be 
bought  for  $9000  ? 

142.  When  gold  is  131,  how  much  of  it  can  be  bought  for 
$360  in  currency  ?     How  much  currency  is  equal  to  $360  in 
gold? 


236  EQUATION  OF  PAYMENTS. 

143.  A  factory  having  been  insured  7  years  for  $21,000,  at 
1%  %>  was  destroyed  by  fire  ?  what  was  the  actual  loss  to 
the  Insurance  Company  ? 

144.  A  certain  town  is  taxed  $2328,  and  pays  3  %  for  col- 
lecting it ;  the  property  is  valued  at  $419,568,  and  there  are 
300  polls  taxed  $1  each  ;  what  is  H.  Scott's  tax,  whose  prop- 
erty is  valued  at  $3100  ? 

145.  For  what  amount  can  a  bill  of  exchange  on  England  be 
bought  for  $3633,635,  at  9  %  premium  ? 


EQUATION  OF  PAYMENTS. 

Art,  169, — Equation  of  Payments  is  finding  the  time 
when  several  payments,  due  at  different  times,  may  be 
made  at  once,  or  the  balance  of  an  account  may  be  paid, 
without  loss  to  either  debtor  or  creditor.  The  time  for 
such  payment  is  called  equated  time. 

An  account  current  is  a  record  of  what  one  person  is 
debtor  or  creditor  to  another. 

CASE  I. 

Art,  170, —  When  the  payments  are  due  at  different  times 
to  find  the  equated  time. 

EXAMPLE  1. — I  owe  J.  Brush  &  Co.  $400,  payable  in  3  mos., 
$300  in  4  mos.,  and  $200  in  6  months  ;  when  should  I  pay  the 
whole  debt  at  once  ? 

Process.— $400  for  3  months  =  $1200  for  1  month. 
$300  for  4       «       =$1200 
$200  for  6       "       =  $1200 
$900  for—     «       =$3600 

Now  $3600  for  1  month  =  $900  for  as  many  {  $900)  $3600 
months  as  it  contains  times  $900.  j          Ans.  4  inos. 

When  the  times  of  different  payments  are  reckoned  from  different 
dates,  any  date  may  be  assumed  from  which  to  reckon  them  and  the 
equated  time.  It  is  more  convenient  to  assume  the  date  of  the  ear- 
liest payment. 


EXAMPLES.  237 

Ex.  2.  What  is  the  equated  time  of  the  following  account  ? 

T.  A.  SCOTT,  Dr. 

To  STJYDAM  &  JACKSON  : 

1860.  Aug.  10.  Mdse.  (3  mos.  credit) $300 

"      Sept.  15,     "       (6  mos.) 

1861.  Jan.  1,  Cash 

Process. — These  different  sums  will  be  due — 

1860.  Nov.  10,  $300  X  0  days  =  $0 

1861.  Mar.  15,  $400X124  "   =$49600  for  1  day. 
"      Jan.  1,      $500  X  51    "   =  $25500 

$1200  X—    "  =)$75100 

Ans.  63  days. 

RULE. — Multiply  each  payment  by  its  time,  and  divide  the 
sum  of  the  products  by  the  sum  of  the  payments. 

This  rule  is  according  to  bank  discount. 

If  the  date  is  required,  reckon  the  equated  time  from  the  given  or 
assumed  date. 

Count  i  day  or  more  as  1  day. 

Cash  payments  have  no  products  when  the  date  is  given,  but  must 
be  included  in  the  sum  of  the  payments. 

EXAMPLES. 

3.  A  merchant  buys  goods,  pays  $200  at  the  time,  and  agrees 
to  pay  $200  in  4  months,  and  $200  in  8  months  ;  what  is  the 
equated  time  of  payment  ? 

4.  A  merchant  bought  goods  on  4  months'  credit,  as  follows  : 
April  10,  $200  ;   May  15,  $160 ;   June  1,  $440 ;   what  is  the 
equated  time  of  payment  ? 

5.  A  man  borrowed  $500,  and  agreed  to  pay  $100  in  2  mos., 
$200  in  4 mos.,  and  the  balance  in  6  mos. ;  what  is  the  equated 
time  ? 

6.  The  following  bills  of  goods  were  bought  on  six  months' 
credit :  Aug.  5,  $62.50  ;  Aug.  11,  $24.50  ;  Sept.  10,  $32.60  ; 
Oct.  15,  $50.     If  a  note  payable  in  6  months  was  given  for  the 
whole  amount,  when  ought  it  to  have  been  dated  ? 

7.  A  man  bought  a  farm  for  $8000,  and  agreed  to  pay  $2000 
at  the  time,  and  the  rest  in  three  annual  payments  ;  what  is 
the  equated  time  of  payment  ? 


238  EQUATION  OF  PAYMENTS. 

8.  The  following  bills  of  goods  have  been  bought  on  60  days 
credit ;  May  1,  $150 ;  June  16,  $200  ;  July  20,  $320  ;  Sept.  1, 
$300  ;  what  is  the  equated  time  of  payment  ? 

9.  A  man  owes  $1600,  %  of  which  is  now  due  ;  %  of  it  in  4 
months  ;  %  in  6  months,  and  the  rest  in  8  months  ;  what  is  the 
equated  time  of  payment  ? 

10.  A  merchant  has  bought  goods  as  follows  :  Aug.  10,  $160 
on  60  days'  credit ;  Sept.  1,  $250,  90  days'  credit;  Oct.  12,  $300, 
60  days  ;  Nov.  10,  $200,  90  days  ;  what  is  the  equated  time  of 
payment  ? 

11.  A  merchant  has  sold  goods  amounting  to  $1200,  of  which 
he  received  $300  at  the  time  ;  $300  was  to  be  paid  in  3  months  ; 
$300  in  6  months,  and  the  remainder  in  9  months  ;  what  is  the 
equated  time  of  payment  ? 

12.  Sold  goods  as  follows  :  Sept.  5,  $500  on  2  months'  credit ; 
Oct.  10,  $400,  3  months ;  Nov.  15,  $600,  3  months  ;  Dec.  1, 
$500,  1  month  ;  what  is  the  equated  time  of  payment  ? 

CASE  H. 

Art.  171* — To  find  ike  equated  time  of  the  balance  of  a 
debt,  when  partial  payments  have  been  made  before  it  is  due. 

Ex.  13. — J.  Howe  owed  me  $500  payable  in  6  mos.,  but  at 
the  end  of  3  months  he  paid  me  $100,  and  at  the  end  of  another 
month  $200  ;  how  long  may  the  balance  remain  unpaid  ? 

Process.— $100  for  3  months  before  due  =  $300  for  1  month. 
$200  for  2  months  before  due  =  $400 

$300  for—     "  "        "  =$700 

Therefore  the  balance  ($500— $300)  $200  may  remain  unpaid  as 
many  months  as  it  is  contained  times  in  $700.  $200)  $700 

Ans.  85  mos. 

EULE. — Multiply  each  partial  payment  by  the  time  it  was 
made  before  it  was  due,  and  divide  the  sum  of  the  products 
by  the  balance  unpaid. 

EXAMPLES. 

14.  J.  King  promised  to  pay  $800  in  10  months.  At  the 
end  of  4  months  he  paid  $200,  and  after  3  more  months  $100 ; 
how  long  may  he  wait  before  paying  the  balance  ? 


EXAMPLES. 


239 


15.  A  merchant  owed  $1000,  payable  in  6  months.     Atrthe 
end  of  two  months  he  paid  $200,  and  at  the  end  of  three  more 
months  $300  ;  how  long  may  he  leave  the  balance  unpaid  ? 

16.  A  man  gave  his  note  for  $1200,  payable  in  6  months  ; 
at  the  end  of  the  first,   third  and  fifth  months  he  paid  $100 
each  time  ;  how  long  may  he  keep  the  balance  ? 

17.  A  man  bought  a  house  for  $2400,  payable  in  2  years,  but 
at  the  end  of  1  year  he  paid  -|  of  it ;  how  long  after  the  whole 
was  due  may  he  wait  before  paying  the  balance  ? 

CASE  HI. 

Art.  172. — To  find,  the  equated  time  of  an  account  cur- 
rent. 

EXAMPLE  18. — 

DR.  J.  H.  MEQLEB.  CB. 


1867. 

1867. 

April     1 

To  Mdse. 

$700 

July    11 

By  Cash 

$200 

"      16 

« 

200 

Aug.      1 

<  : 

300 

May    11 

it 

100 

Sept.   21 

« 

100 

June    16 

Cash 

400 

Oct.       1 

Mdse. 

200 

Process. — First  find  the  equated  time  of  each  side. 


Dr.  $700X0  days  = 

200X15  "    =$3000  for  Id. 
100X41  "    =  4100     " 
400X77  "    =30800     " 
1400 


)37900     " 
From  April  1,  27  days. 
Dr.  April  28,  $1400. 


Or.  $200X0  days  = 

300X20"    =$6000  for  Id. 
100X72  "    =    7200      " 
200X82  "    =  16400      " 
$800 


) $29600      " 
From  July  11,  37 
Cr.  Aug.  17,  $800. 

Difference  between  April  28and  August  17  =  111  days. 
If  the  account  is  balanced  April  28,  credit  is  given  for  $800,  111 
days  before  it  is  paid,  which  would  be  a  loss  to  the  creditor.  There- 
fore the  balance,  ($1400 — $800)  $000,  is  due  as  many  days  before 
April  28  as  it  is  contained  times  in  ($800X111  days)  $88800,  or  148 
days.  Hence  the  balance  was  due  Dec.  2,  1866. 

RULE. — Find  the  equated  time  of  each  side ;  multiply  the 
amount  of  the  smaller  side  by  the  number  of  days  between  the 
two  dates  of  equated  time,  and  divide  the  product  by  the  bal- 
ance ;  the  quotient  will  be  the  number  of  days  to  be  ADDED  to 


240 


EQUATION   OF  PAYMENTS. 


the  equated  time  of  the  larger  side  when  its  amount  becomes 
duei;A.ST,  but  SUBTRACTED  from  it  when  it  becomes  due  first. 

The  cash  value,  or  true  balance,  at  any  time  of  settlement,  is  found 
by  adding  the  interest  up  to  the  time  of  settlement,  when  the  balance 
is  due  beforehand,  and  subtracting  it  when  due  afterwards. 

EXAMPLES. 

19.  What  must  be  the  date  of  a  note  to  balance  the  follow- 
ing account  ? 
DR.  J.  WILSON.  CB. 


1867. 

1867. 

May      1 

To  Mdse. 

$900 

April     1 

By  Cash 

$200 

"      16 

« 

700 

"      16 

« 

400 

June     1 

Mdse. 

400 

May    15 

Draft 

500 

20.  What  must  be  the  date  of  a  note  to  balance  the  follow- 
ing account  ? 

DB.  G.  S.  WOOD.  CB. 


1866. 

1866. 

March  1 

To  Mdse.  (4m.) 

$1000 

June   16 

By  Cash 

$500 

April  10 

"       (3m.) 

800 

July    10 

« 

400 

June    11 

« 

600 

« 

Draft  (10  d.) 

600 

CASE  IV. 

Art.  173. — To  find  the  true  balance  of  an  account  bearing 
interest  when  the  time  of  settlement  is  given. 

EXAMPLE  21. — What  is  the  true  balance  of  the  following  ac- 
count, at  the  time  of  settlement,  August  20,  1868,  allowing  60 
days'  credit  to  each  charge,  and  7  per  cent,  interest  ? 

DB.  THOS.  GOODWIN.  CB. 


1868. 

1 

1868. 

Jan.       2 

To  Mdse. 

$200 

1  Feb.     20 

By  Mdse. 

$100 

April  20 

« 

400 

May     10 

« 

300 

REDUCTION   OF   CURRENCIES. 


241 


Dr.     Due  Mar.  2,  $200. 
Int.  till  Aug.  20,  AM'T. 

(172  days).  . . .       6.60  $206.61 

Due  June  20 $400 

Int.  till  Aug.  20, 

(61  days) 4.66     404.66 


$611.27 
404.69 


Or.    Due  April  20,  $102. 
Int.  till  Aug.  20,  AMT. 

(121  days)  . . . .     $2.36   $102.32 

Due  July  10 300 

Int.  till  Aug.  20, 

(41  days) 2.37     302.37 

$404.69 


True  balance,  Aug.  20 . .  $206.58 

EULE. — Find  the  interest  on  each  entry  up  to  the  time  of 
settlement.  Add  the  several  amounts  on  each  side,  and  the 
difference  between  the  sums  will  be  the  true  balance. 

EXAMPLE  22. 

DK.  HOWE  &  STEBBINS.  OB. 


1863. 

1863. 

July      1 

To  Mdse. 

$250 

July    21 

By  Consignm't 

$130 

Aug.    13 

« 

200 

Oct.       1 

Draft 

160 

Sept.  24 

« 

550 

10 

n 

210 

What  is  the  balance  due  Jan.  1,  1864,  allowing  6%  interest? 


REDUCTION  OP  CURRENCIES, 
Art,  174,— Reduction  of  Currencies  is  changing  one 
kind  of  Money  into  another  without  altering  its  value. 

State  Currencies. — Sterling  Money  was  formerly  the 
currency  of  this  country  before  its  separation  from  Eng- 
land, and  is  still  used  to  some  extent.  But  its  value  is 
not  the  same  as  in  England ;  nor  the  same  in  all  the 
States,  because  its  paper  bills  depreciated  more  in  some 
than  in  others.  Hence  arose  the  following  currencies  : 

N.  E,  States, 
Virginia, 
Kentucky, 
Tennessee, 


N.  E.  Currency  in 


11 


242  SEDUCTION  OF  CURRENCIES. 

(  New  York,        ) 

N.  Y.  Currency  in    -<  Ohio,  V  $1  =  8s.  =  ££ . 

(  N.  Carolina,     ) 

f  Pennsylvania, 


Penn.  Currency  in        g^SST 

[   Maryland, 


,  =  7s.  6d.=£f. 
Maryland, 

Geor.  Currency  in   j    g^afdina        '  $1=4s-  8d-  = 
Hence,  also,  in 
New  England  Currency.  .£1=$^=$3. 33^  ;  ls.=16f  cts. 

New  York  Currency £l=$f  =$2.50    ;  Is.  =12^   " 

Pennsylvania  Currency.. £!=$§  =$2.66|  j  Is.^l3|   '• 
Georgia  Currency £l=$^=$4.28f  ;  ls.=22|    " 

Art.  175. — To  reduce  U.  S.  Money  to  State  Currencies. 

EXAMPLE  1. — What  is  the  value  of  $836.50  in  Pennsylvania 
currency  ? 


$836.50 


Process.  —  $1—  (7s.  6d.)  7£s.,  or  £|  ;  there- 
fore  $836.50:=7i  times  as  many  shillings,  or 
|  times  as  many  pounds. 


20)6273.  75  i 


Ans.  £313.13s.  9.00d. 

RULE.  —  Multiply  the  given  sum  by  the  value  of  $1  in  the 
required  currency. 

EXAMPLES. 

2.  Reduce  $315.4375  to  New  York  currency. 

3.  Reduce  $490.  38  to  New  England  currency. 

4.  Reduce  $325.00  to  Pennsylvania  currency. 

5.  Reduce  $245.00  to  Georgia  currency. 

Art.  176.  —  To  reduce  State  Currencies  to  U.  S.  Money. 

EXAMPLE  6.—  What  is  the  value  of  £312  18s.  9d.  ? 

£312  18s.9d.=.75s. 

Process.  —  7£s  .=$1,  therefore  £312  20 

18s.  9d.  reduced  to  shillings,  and  di-  7^6258  75 

videdby7S=$834.50. 


EXAMPLES. 


243 


EULE. — Divide  the  given  sum  reduced  to  shillings,  by  the 
number  of  shillings  in  $1  ;  or  reduced  to  pounds,  by  the  frac- 
tion of  a  pound,  equal  to  $1. 

EXAMPLES. 

7.  Reduce  £90  6s.  New  York  currency  to  U.  S.  money. 

8.  Eeduce  £120  7s.   6d.   Pennsylvania  currency  to  U.    S. 
money. 

9.  Eeduce  £35  14e$.  9d.   New  England  currency  to  U.   S. 
money. 

10.  Eeduce  £57  12s.  6d.  Georgia  currency  to  U.  S.  money. 

Art.  177. — TABLE  OF  THE  PRINCIPAL  FOREIGN   COINS  AND 
THEIR  VALUE  IN  U.  S.  MONEY. 


Austria,  Florin  of  
Canada,  Pound  Ster.. 
China  Tael  

$0.48^ 
4.00 
1  48 

Italy,  Dollar  of  Eome, 
'  '    Ducat  of  Naples, 
"    Lira  of  Sardinia 

$1.05 
.80 
ISA 

Denmark,  Dol.,  (sp.). 
England,  Pound  Ster. 
*  *         Crown 

1.05 

4.84 
1  06 

'  '    Lira  of  Tuscany, 
"    Livre  of  Genoa, 
Mexico,  Doubloon 

••*-wio 
.16 

.ISA 

15  60 

.18  A 

Portugal,  Milrea  .... 

1.12 

**          LlVT6 

18 

Prussia    Florin  

22  ^ 

Germany  Florin 

40 

Eussia   Euble 

•**  74 

75 

Eix  Dollar, 
India  Pagoda  

.69 
1  94 

Spain,  Eeal  Plate  
Switzerland,  Livre  .  .  . 

.10 

27 

"      Eupee  

44^o 

To  reduce  Sterling  Money  of  England  to  U.  S.  Money,  multiply 
pounds  and  decimal  of  a  pound  by  $4.84.  To  reduce  U.  S.  Money 
to  pounds  sterling  divide  by  4. 84 .  Eeduce  the  decimal  to  shillings, 
&c. 

EXAMPLES. 

11.  Eeduce  1500  francs  to  U.  S.  Money. 

12.  Eeduce  3000  livres  of  Switzerland  to  U.  S.  Money. 

13.  Eeduce  £140  15s.  9d.  to  U.  S.  Money. 

14.  Eeduce  100  dollars  (specie)  of  Denmark,  to  U.  S.  Money. 

15.  Eeduce  500  rupees  of  India  to  U.  S.  Money. 

16.  Eeduce  400  rubles  of  Eussia  to  U.  S.  Money. 

17.  Eeduce  $705.6115  to  Pounds  Sterling,  &c. 


BATIO. 

Promiscuous  Examples, 

1.  Beduce  £89  18s.  N.  York  Currency  to  U.  S.  Money. 

2.  Beduce  $4S8.3S  to  New  England  Currency. 

3.  Beduce  £36  9d.  N.  England  Currency  to  U.  S.  Money. 

4.  Beduce  $314.43%  to  New  York  Currency. 

5.  Beduce  500  francs  to  U.  S.  Money. 

6.  Beduce  $810  to  Swiss  livres. 

7.  Beduce  £60  15s.  Pennsylvania  Currency  to  U.  S.  Money. 

8.  Beduce  $417.25  to  Pennsylvania  Currency. 

9.  Beduce  400  India  rupees  to  U.  S.  Money. 

10.  Beduce  £114  16s.  Georgia  Currency  to  U.  S.  Money. 

11.  Beduce  $954.12^  to  Georgia  Currency. 


RATIO. 

Artt  178. — Ratio  denotes  the  magnitude  of  one  number 
compared  with  another  of  the  same  kind. 

Arithmetical  ratio  denotes  the  difference  between  two 
numbers  ;  geometrical  ratio  the  number  of  times  one 
contains  the  other.  The  word  ratio,  used  alone,  means 
the  latter. 

Ratio  equals  the  quotient  of  one  number  divided  by 
another,  and  is  usually  expressed  by  (  :  )  written  be- 
tween them  ;  thus  the  ratio  of  12  to  3  is  written  12  :  3. 

The  two  numbers  are  called  the  terms  of  the  ratio ; 
written  together  they  are  called  a  couplet,  of  which  the 
first  is  called  the  antecedent,  and  the  second  the  conse- 
quent. 

Ratio  respects  only  things  of  the  same  name,  or  which  may  be  re- 
duced to  the  same  ;  thus  the  ratio  of  12  inches,  or  1  foot  to  6  inches 
is  2,  but  there  is  no  ratio  .between  12  inches  or  1  foot  and  6  cents. 

Art.  179.  -A  Compound  ratio  is  the  product  of  two  or 
more  simple  ratios ;  thus  the  ratio  of  2:6.  and  3  :  9. 
make  the  compound  ratio  2  X  3  :  6  X  9>  or  6  :  ^ 


PROPORTION.  245 


EXERCISES. 


Express  the  ratios  of  4  to  8,  3  to  9,  5  to  15,  7  to  14.  12  to  24. 
What  is  the  ratio  of 


7:2:12 
6:24 

40:10 
5:25 


$4 :  $12 
10  Ib.  :  30  Ib. 
12  gal.  :  3  gal. 
8  rods  :  32  rods 


50  Ibs.  :  1  cwt. 
3  qrs.  :  4  yds. 
24  qts.  :  3  gals. 
£1 :  2s.  6d. 


A  ratio  may  be  reduced  to  its  lowest  terms  by  dividing  both  terms 
by  their  greatest  common  divisor  ;  thus,  5 :  25  =  1 :  5. 


PROPORTION. 

Art.  180. — Proportion  is  an  equality  of  ratios.  The 
ratio  of  4  :  2  =  12  :  6  ;  hence  4:2  is  in  proportion  as 
12:  6. 

Proportion  is  usually  expressed  by  (::)  four  dots 
placed  between  two  ratios ;  thus,  6  :  3  : :  24  :  12  ;  which 
is  read  6  is  to  3  as  24  to  12.  The  first  and  last  terms 
are  called  the  extremes  ;  the  second  and  third  the  means. 

When  the  consequent  of  the  first  ratio  is  the  same  as 
the  antecedent  of  the  next,  it  is  called  a  mean  proportion- 
al ;  thus,  8  :  4  : :  4  :  2  ;  4  is  a  mean  proportional. 

Art.  181. — Proportion  is  often  the  most  convenient 
method  of  solving  arithmetical  questions,  when  the  re- 
quired number  is  evidently  as  many  times  greater  or 
less  than  another  of  the  same  kind,  as  is  expressed  by 
the  ratio  of  two  other  numbers  on  which  they  respect- 
ively depend ;  thus, 

Prices  depending  on  quantities  ; 

Times,  &c.,  depending  on  distances,  &c., 
are  evidently  in  proportion. 

1  Ib.  of  coffee  :  10  Ibs.  of  coffee  ::  $0.45  :  $4.50  ; 
12  men:  4  men:: 6  days:  2  days  (in  doing  a  certain  work.) 


246  PEOPOBTION. 

The  proportion  is  called  direct  when  a  greater  number  of  one  kind 
requires  a  greater  number  of  another  kind,  or  less  requires  less  ;  but 
indirect  or  inverse  when  a  greater  number  requires  a  less,  or  a  less 
requires  a  greater  ;  thus  the  first  proportion  above  is  direct,  because 
the  greater  quantity  (10  Ibs.)  requires  the  greater  price  (34  50  ;)  but 
the  other  is  inverse,  because  the  greater  number  of  men  (12)  requires 
the  less  number  of  days  (2,)  to  do  a  certain  work. 

In  every  proportion  the  product  of  the  extremes  is 
equal  to  the  product  of  the  means ;  thus,  3  :  4  : :  6  :  8, 
3X8  =  4X6.  Hence, 

The  product  of  the  means  divided  by  either  extreme 
gives  the  other  ;  and 

The  product  of  the  extremes  divided  by  either  mean 
gives  the  other. 

EXAMPLE  I.— If.  5  yards  of  linen  cost  $6.00,  what  will  20 
yards  of  linen  cost  ? 

Process. — This  is  an  example  in  proportion,  because  it  is  evident 
that  the  prices  must  be  in  proportion  to  the  given  numbers  of  yards. 
Since,  too,  the  price  of  20  yards  is  unknown,  it  is  usually  considered 
as  the  fourth  term  of  the  proportion,  which  is  to  be  found,  and  there- 
fore the  corresponding  number  or  given  price  will  be  the  third  term. 
It  is  evident,  also,  that  20  yards  will  cost  more  than  5  yards  ;  there- 
fore, 20  yards  must  be  the  second  term,  and  5  yards  the  first  term,  in 
order  to  make  the  ratios  equal,  or  a  proportion.  Hence, 
5  yards  :  20  : :  $6 :  Ans. 

Beducing  the  first  ratio  to  its  lowest  terms, 

1  yard  :  4  yards  : :  $6  :  Ans.  =  $6X*_  $24 

BULK. — Consider  the  answer  the  fourth  term,  and  make  the 
given  number  of  the  same  name  or  kind  the  third  term  ;  then, 
if  the  answer  will  evidently  be  greater  than  the  third  term, 
make  the  greater  of  the  other  two  numbers  the  second  term  ; 
but,  if  less,  make  the  smaller  number  the  second  term.  The 
remaining  number  will  be  the  first  term. 

Reduce  the  left  hand  ratio  to  its  lowest  terms,  or  cancel  any 
factor  common  to  the  first  lerm  and  either  the  second  or  third. 
Then  multiply  the  second  and  third  terms  together,  and  di- 
vide by  the  first. 


EXAMPLE&  247 

Compound  Numbers  must  be  reduced  to  the  lowest  denomination 
mentioned,  and  the  first  and  second  terms  must  be  of  the  same  name 
as  well  as  kind. 

EXAMPLES. 

2.  If  5  yards  of  cloth  cost  $35,  how  much  will  20  yds.  cost  ? 

3.  If  12  tons  of  coal  cost  $72,  how  much  will  3  tons  cost  ? 

4.  If  7  Ibs.  of  coffee  cost  $2.33)^,  how  much  will  4  Ibs.  cost  ? 

5.  If  3  bbls.  of  flour  cost  $22.50,  how  much  will  50  barrels 
cost? 

6.  If  44  Ibs.  of  tea  cost  $33.00,  how  much  wiU  11  Ibs.  cost  ? 

7.  If  10  acres  of  land  produce  250  bushels  of  corn,  how 
much  will  45  acres  produce  ? 

8.  If  a  man  travel  300  miles  in  12  days,  how  far  can  he 
travel  in  4  days  ? 

9.  If  1  Ib.  6  oz.  of  silver  is  worth  $12.50,  what  are  3  oz.  10 
pwts.  worth? 

10.  If  124  men  can  build  a  mill-dam  in  60  days,  in  how 
many  days  could  248  men  build  it  ? 

11.  If  a  man  can  walk  3%  miles  in  1  hour,  how  long  will  it 
take  him  to  walk  12  miles  120  rods  ? 

12.  How  many  yards  of  cambric,  %  yd.  wide,  will  it  take  to 
line  20  yards  of  cloth,  1%  yds.  wide  ? 

13.  If  a  quantity  of  provisions  will  last  315  men  56  days, 
how  many  days  will  the  same  last  45  men  ? 

14.  If  a  quantity  of  provisions  will  last  316  men  56  days, 
how  many  men  will  it  feed  14  days  ? 

15.  If  %  of  a  yard  of  cloth  cost  $1\,  what  will  3%  yds.  cost  ? 

Art.  182. — Proportion  may  be  used  in  solving  ques- 
tions under  many  of  the  preceding  rules. 

16.  Percentage.— What  is  7  %  of  $256  ? 

By  Proportion,  $100  :  $256  : :  7  %  :  $17.92,  Ans. 

17.  What  per  cent,  of  $400  is  $24  ? 

$400:  $100::  $24:6,%,  Ans. 

18.  Problems  in  Interest— How  long  must  $200  be  at  6  %  in- 
terest to  gain  $36  ? 

Interest  1  year  is  12,  therefore  $12  :  $36  : 1  year :  3  yrs. ,  Ans. 


248  PROPORTION. 

19.  If  the  interest  of  $600  for  1  year  8  mos.  is  $60,  what  is 
the  rate  per  cent.? 

Interest  at  1  %  is  $10,  therefore  10  :  60  : :  1 :  6  %,  Ans. 

20.  What  principal  at  6  per  cent,  will  yield  $4.52  interest  in 
1  year  4  months  be  ? 

Interest  of  $1  is  8  cents,  therefore  .08  :  4.52  : :  $1 :  $56.50,  Ans. 

21.  Discount. — What  is  the  present  worth  and  discount  of 
$306,  due  in  4  mos.,  at  6  %  discount  ? 

The  present  worth  of  $1.02  is  $1.00,  therefore  $1.02  : 100  : :  $306  : 
$300,  Ans. 

The  discount  of  $1.02  is  $.02,  therefore  $1.02  :  $.02  : :  $306:  $6, 
Ans. 

22.  Profit  and  Loss. — Sold  a  horse   for   $150,  and  gained 
25  %  ;  what  did  the  horse  cost  ? 

$125  :  $100  : :  $150  :  $120,  Ans. 

23.  Partnership. — Messrs.  Platt,  Wood  &  Torrey  are  in  part- 
nership ;  P.'s  capital  is  $5000,  W.'s  $4000,  T.'s  $3000;  they 
have  gained  $3600  ;  what  is  each  one's  share  ? 

The  whole  capital  $12,000 :  $5000 : :  $3600  :  $1500,  P.'s  share,  &c. 

Solve  the  f ollowing  questions  by  proportion . 

24.  What  is  6%  of  $750? 

25.  What  per  cent,  of  $500  is  $60  ? 

26.  What  is  $650  in  currency  worth  when  gold  is  130  ? 

27.  What  is  the  interest  of  $150  for  4  years  2  mos.,  at  6  %  ? 

28.  What  principal,  at  6  %,  will  gain  $60  in  4  years  ? 

29.  What  is  the  present  worth  of  $1350,  due  in  5  years  10 
mos.,  at  6%? 

30.  What  was  the  cost  of  a  yard  of  cloth,  which  being  sold 
for  $4.25  occasioned  a  loss  of  20  per  cent.  ? 

31.  A  bankrupt's  debts  amount  to  $9600,  and  he  has  proper- 
ty amounting  to  $7500 ;  how  much  can  he  pay  a  creditor  whom 
he  owes  $1600  ? 

32.  Messrs.  Bay  &  Stearns  are  in  partnership  ;  B.  's  capital 
is  $3000,  S.'s  capital  $2000  ;  they  have  gained  $2500  ;  what  is 
each  one's  share  ? 

33L  For  what  amount  must  a  note  for  sixty  days  be  given,  to 
obtain  from  a  bank  $1800,  at  6  %  discount  ? 


COMPOUND   PROPORTION.  249 

34.  What  is  the  interest  of  $225  for  2  years  7  mos.,  at  7  per 
cent.? 


COMPOUND  PROPORTION. 

Art.  183.— Compound  Proportion  is  the  equality  of  a 
compound  and  a  simple  ratio.  It  consists  of  two  or 
more  simple  proportions. 

EXAMPLE  1. — If  7  men  can  cut  42  acres  of  wheat  in  3  days, 
working  10  hours  a  day,  how  many  acres  can  14  men  cut  in  4 
days,  working  9  hours  a  day  ? 

Process. — Since  the  answer  will  be  acres,  42  acres  will  be  the  third 
term.  The  other  numbers,  two  of  the  same  kind  make  the  first  and 
second  terms  of  as  many  simple  proportions,  each  couplet  to  be  ar- 
ranged as  if  the  answer  'depended  upon  it  alone  ;  thus, 

%  523 

H  :  U  :  :  42       $  X  3  X*0)±2  X .* 
$  :    4  by  cancellation. 

5  *0:    03  5)  (42X^X3)504 

Ans.  lOOf   acres. 

RULE. — Make  that  number  which  is  of  the  same  name  as 
the  answer  the  third  term.  Arrange*  the  other  numbers  in 
pairs  of  the  same  kind,  as  the  first  and  second  terms  of  as 
many  simple  proportions  as  there  are  pairs,  and  each  couplet 
as  if  the  answer  depended  on  it  alone. 

Cancel  as  in  simple  proportion,  or  reduce  the  ratios  and 
divide  the  product  of  all  the  second  and  third  terms  by  the 
product  of  att  the  first  terms. 

EXAMPLES. 

2.  If  6  men  can  dig  a  ditch  36  rods  long  in  8  days,  how 
many  men  will  it  require  to  dig  a  ditch  72  rods  long  in  4 
days? 

3.  If  90  Ibs.  of  beef  will  supply  12  men  20  days,  how  long 
will  144  pounds  last  36  men  ? 

11* 


250  CONJOINED   PROPORTION. 

4.  If  6  horses  eat  48  busliels  of  oats  in  12  days,  how  many 
horses  will  eat  96  bushels  in  8  days  ? 

5.  If  <?100  gain  $7  in  12  months,  how  long  will  it  take  $600 
to  gain  $21  ? 

6.  If  6  men  can  dig  4  acres  of  potatoes  in  10  days,  in  how 
many  days  will  18  men  dig  24  acres  ? 

7.  If  8  men  can  dig  a  cellar  40  feet  long,  27  feet  wide,  and 
8  feet  deep,  in  12  days,  how  many  men  will  dig  a  cellar  50  feet 
long,  36  feet  wide  and  9  feet  deep  in  the  same  time  ? 


CONJOINED  PROPORTION. 

Art,  184.  —  Conjoined  Proportion  is  a  proportion  in 
which  each  antecedent  of  a  compound  ratio  is  equal  in 
value  to  its  consequent. 

EXAMPLE.  —  If  10  bushels  of  corn  will  pay  for  5  loads  of 
wood,  and  20  loads  of  wood  for  4  tons  of  hay,  how  many  bush- 
els of  corn  will  pay  for  25  tons  of  hay  ? 

Process.  —  Since  10  bushels  corn  =^5  loads  of  wood,  20  loads  of 
wood  =  *£•  of  10  bushels  corn  ;  and  since  it  also  equals  4  tons  of 
hay,  25  tons  of  hay  =  *£  of  a5a  of  10  bushels  corn  ==  Ana.  250  bush. 

10  bushels  corn  =  5  loads  of  wood,       2        5 

20  loads  wood    =  4  tons  of  hay,         10X20X25 

25  tons  of  hay  = 


RULE.  —  Write  the  equal  quantities,  with  the  sign  of  equality 
between  them,  under  one  another,  in  pairs,  so  that  each  con- 
sequent shall  be  of  the  same  name  as  the  next  antecedent, 
placing  the  odd  quantity  on  the  opposite  side  of  that  of  the 
same  name. 

Cancel  the  factors  common  to  both  sides,  and  divide  the 
product  of  all  the  quantities,  on  the  side  containing  the  odd 
one,  by  the  product  of  all  on  the  other  side. 

Art,  185,  —  Arbitration  of  Exchange  is  finding  the 
rate  of  exchange  between  two  countries,  through  inter- 


ALLIGATION.  .         251 

^^     r>         ^ 

mediate  exchanges   between   other   cou&teafefik.-  '^SMs  is 
done  by  Conjoined  Proportion. 

EXAMPLES. 

2.  If  $9  in  the  United  States  are  equal  to  12  rubles  at  St. 
Petersburg,  and  8  rubles  in  St.  Petersburg  are  equal  to  15 
florins  in  Frankfort,  and  9  florins  in  Frankfort  are  equal  to  20 
francs  in  Paris,  how  many  dollars  in  the  United  States  are 
equal  to  75  francs  in  Paris  ? 

3.  If  25  yards  of  cloth  are  worth  22  barrels  of  flour,  11  bbls. 
of  flour  are  worth  150  pounds  of  wool,  and  15  Ibs.  of  wool  are 
worth  18  Ibs.  of  butter,  how  many  pounds  of  butter  will  pay 
for  10  yards  of  cloth  ? 


ALLIGATION. 

Art.  186. — Alligation  is  finding  the  prices  or  quantities 
of  mixtures.  It  is  Medial  or  Alternate. 

The  word  alligation  means  connecting  together,  and  is  used  in 
arithmetic  because  the  prices  of  mixtures  are  connected  one  with 
another. 

Art.  187.— Alligation  Medial  is  finding  the  price  of  a 
mixture  when  the  quantity  and  price  of  each  ingredient 
are  given. 

EXAMPLE  1.— A  farmer  has  mixed  20  bushels  of  corn,  at 
75  cents,  30  bu.  barley  at  60  cts.,  40  bu.  oats  at  50,  and  10  bu. 
rye  at  $1 ;  what  is  a  bushel  of  the  mixture  worth  ? 

Process.  —20  bushels  corn    X  $.  75  —  $15. 00 

30      •«       barley  X     60=    18.00 

40      "       oats     X     50=    20.00 

_10      «       rye      Xl-00=    10.00 

100  bu.  of  mixture  =   53.00 

1  =53  cts.,  Ans. 

RULE. —  Divide  the  price  of  the  whole  mixture  by  the 
whole  quantity. 
Ex.  2.  A  grocer  mixed  8  Ibs.  of  tea  at  75  cents,  12  Ibs.  at  60 


6 
16 


252  ALLIGATION. 

cts.,  15  Ibs.  at  50  cts.,  and  20  Ibs.  at  40  cts. ;  what  was  a  pound 
of  the  mixture  worth  ? 

Art,  188, — Alligation  Alternate  is  finding  the  quanti- 
ties of  different  ingredients  whose  prices  are  known,  to 
form  a  mixture  worth  a  certain  price. 

Ex.  3.  A  grocer  wishes  to  make  a  mixture  of  tea  worth  56  cts. 
He  has  one  kind  at  40  ots.  a  pound,  another  50  cts.,  another 
60  cts.,  and  another  75  cts. ;  how  much  of  each  may  he  take  ? 

Process.  —  Since  the  gain  and  loss  f40 19 

must  be  equal,  we  connect  a  less  price  j  50 4 

than  that  of   the  mixture  with    one  56  j 

greater.     On  every  pound,  at  40  cents, 

there  is  gained  16  cts.,  and  on  every  [75 

pound  at  75  cts.  there  is  lost  19  cents. 

Therefore,  since  the  gain  is  to  the  loss    The  whole  mixtore=40  Ibs. 

as  16  to  19,  the  quantities  must  be  as 

19  to  16.     For  the  same  reason  the  other  ingredients  at  50  and  60 

must  be  as  4  to  6.     Hence  the  mixture  must  consist  of  19  Ibs.  at  40 

cents,  4  Ibs.  at  50  cts.,  6  Ibs.  at  60  cts.,  and  16  Ibs.  at  75  cts. 

These  relative  quantities  are  found  by  writing  the  difference  be- 
tween the  price  of  the  mixture  and  that  of  each  ingredient  opposite 
the  one  with  which  it  is  connected. 

Art,  189, — The  quantity  of  one  of  the  ingredients  is  some- 
times given. 

In  the  above  example,  let  the  quantity  at  75  cts.  be  8  Ibs.  Then 
since  8  is  only  4  of  16,  only  £  of  the  other  quantities  must  be  taken 
to  form  the  mixture. 

Art,  190, — Again,  the  quantity  of  the  whole  mixture  may 
be  given. 

In  the  same  example,  let  the  quantity  of  the  mixture  be  100  Ibs.  in- 
stead of  40,  as  found.  Then,  since  100  =2^  times  40,  the  quantity 
of  each  ingredient  as  first  found  must  be  multiplied  by  2£. 

EULE. —  Write  the  prices  under  one  another,  and  the  mean 
price  on  the  left.  Connect  each  price  less  than  that  of  tJ.-e. 
mixture  with  one  greater,  and  each  greater  with  one  less. 
Write  the  difference  between  the  price  of  the  mixture  and  that 
of  each  ingredient  opposite  the  price  with  which  the  latter  is 
connected.  The  relative  quantity  of  each  ingredient  will  thus 
be  found  opposite  its  price. 


ALLIGATION.  253 

If  one  of  ike  quantities,  or  the  whole  quantity,  is  given,  and 
is  greater  or  less  than  that  found,  all  the  others  must  be  in- 
creased or  diminished  in  the  same  proportion. 

EXAMPLES. 

Ex.  4.  A  grocer  mixed  different  kinds  of  sugar  at  10,  13  and 
16  cents  a  pound,  so  that  he  could  sell  the  mixture  for  12  cts. 
a  pound  ;  how  much  of  each  kind  did  he  take  ? 

5.  A  farmer  mixed  10  bushels  of  wheat  at  $1.40,  with  rye  at 
96  cents,  corn  at  72  cts.,  and  oats  at  60  cts.,  so  that  he  could 
sell  the  mixture  at  76  cts.  a  bushel ;  how  much  of  each  did  he 
take? 

6.  How  much  water  must  be  mixed  with  wine,  at  90  cents  a 
gallon,  so  that  there  may  be  100  gallons  worth  60  cts.  a  gal.  ? 

7.  A  grocer  has  different  kinds  of  sugar  at  12,  11,  9,  and  8 
oents  a  pound  ;  how  may  he  *nix  them  so  that  he  can  sell  the 
mixture  at  10  cents  a  pound  ? 

8.  How  many  bushels  of  corn  at  $1,  and  oats  at  60  cents  a 
bushel,  must  be  mixed  with  20  bushels  of  rye  at  $1.30,  to  make 
the  mixture  worth  82  cents  ? 

9.  A  wine  merchant  mixed  different  kinds  of  wine  at  62)^, 
87%  and  112}£  cents  a  gallcn,  with  water,  so  that  the  mixture 
was  worth  75  cts. ;  how  much  of  each  did  he  take  ? 

10.  A  grocer  having  sugar  at  8,  16  and  24  cents  a  pound, 
made  a  mixture  of  240  Ibs.  worth  20  cts.  a  pound  ;  how  many 
pounds  of  each  did  he  take  ? 


INVOLUTION. 

Art.  191. — Involution  is  multiplying  a  number  by  itself. 
The  product  is  called  its  power. 

3X3  =  9;  3X3X3  =  27;  9  and  27  are  powers  of  3. 

The  different  powers   are   distinguished  as  the  first, 
second,  third,  &c. ;  and  are  expressed  by  small  figures 


254  INVOLUTION. 

above  the  numbers  at  the  right,  called  an  index  (sing.) 
indices  (plu.,)  the  index  of  the  first  power  being  usually 
omitted. 

The  first       power  of  3  is  written  3 

The  second  power  of  3  is  written  32=  3  X  3 

The  third    power  of  3  is  written  33=  3X3X3 

The  second  power  is  usually  called  the  Square;  the 
third  power  the  Cube. 

KULE. — To  involve  a  number,  or  find  any  power  of  it,  mul- 
tiply the  number  by  itself  one  less  times  than  the  name  of  the 
power  denotes. 

There  is  one  less  multiplication  than  the  times  the  number  is  used 
as  a  factor.  In  3X3=9,  there  is  one  multiplication,  but  3  is  used 
twice  as  a  factor. 

Powers  already  found,  when  multiplied  together,  produce  the 
power  denoted  by  the  sum  of  their  indices ;  32  X^3  or  9X27  = 
36  or  243. 

EXAMPLES. 

What  are  the  squares  or  second  powers  of  1,  3,  5,  7,  9,  11, 
13,  25  ? 

What  are  the  cubes  or  third  powers  of  2,  4,  6,  8,  10,  12  ? 

What  are  the  squares,  cubes,  and  fourth  powers  of  14,  20, 
27,  36  ? 

What  are  the  squares  and  cubes  of  J,  |,  f ,  f,  -^,  .5, 1.2,  .05, 
1.02? 


EVOLUTION, 

Art.  192.— Evolution,  the  opposite  of  Involution,  is 
finding  a  number  which,  multiplied  into  itself,  will  pro- 
duce the  given  number.  The  number  found  is  called  the 
Root  of  the  other,  and  is  one  of  its  equal  factors : 

9  =  3X3;  27  =  3X3X3;  3  is  the  root  of  9  or  27. 


EVOLUTION.  255 

The  root  of  a  square  is  called  the  square  root  ;  of 
a  cube,  the  cube  root  ;  of  a  fourth  power,  the  fourth 
root,  &c. 

The  Radical  Sign,  +/  is  used  to  express  roots.  When 
written  alone  before  a  number  it  expresses  the  square 
root  ;  with  the  figure  3  over  it,  the  cube  root,  &c. 

^/9  expresses  the  square  root  of  9  =  3. 
4/27  expresses  the  cube  root  of  27  =  3. 
expresses  the  fourth  root  of  81  =  3. 


Square  Root. 

Art.  193.  —  The  square  root  of  a  number,  is  another 
number,  which  multiplied  by  itself  once  will  produce  the 
given  number  or  square. 

The  square  of  1=    1  therefore  the    -v/I—  1 

"          "           2=     4=  "  </1=  2 

"          "           3=    9  "  ^9=  3 

4=  16  "  «/16=  ± 

5=  25  "  x/25=  5 

6=  36  "  <v/36=  6 

«          «           7=  49  "  <v/49=  7 

8=  64  "  -v/64=  8 

"         "          9=  81  "  <v^??=  9 

"          "         10=100  "  ^100=10 

*{         11=121  <{  ^121=11 

"         "         12=144  ««  ^144=12 

To  find  the  square  root  of  any  large  number  we  ob- 

serve — 

First  —  That  in  squaring  a  number,  each  figure  is  squared  and  mul- 
tiplied twice  into  each  of  the  others. 

Secondly  —  That  the  square  of  any  number  contains  as  many  pe- 
riods of  two  figures  each,  beginning  at  the  right,  as  there  are  figures 
in  the  root,  except  the  left-hand  period  may  have  only  one  figure. 


256  EVOLUTION. 

Thirdly  —  That  tlie  square  of  a  unit's  figure  is  in  the  corresponding 
unit's  period  ;  of  a  ten's  figure,  in  the  ten's  period  ;  of  a  hundred's 
figure,  in  the  hundred's  period,  &c.  ;  also,  that  the  square  of  the  left- 
hand  figure  is  the  greatest  square  in  its  corresponding  period. 

The  square  of  any  number  from 

1  to        9  is  from  1  to  81  and  lias  1  period. 

10  to      99  is  from          I'OO  to  98  01  and  has  2  periods. 

100  to    999  is  from      10000  to       99'80  01  and  has  3  periods. 

1000  to  9999  is  from  I'OO'OO'OO  to  99'9800'01  and  has  4  periods. 

Therefore  the  square  root  of  any  number  from 

1  to  '81  is  from      1   to      9.    1   figure. 

I'OO  to        98'01   is  from     10  to    99.    2  figures. 
10000  to  9980'01  is  from   100  to  999.    3  figures. 

Or  we  may  consider  a  square  number  as  representing 
some  square  surface,  as  a  floor  or  field,  &c. 
The  square  of  36  is 


or 


To  find  the  square  root  of  1296  we  reverse  this  pro- 
cess : 

Dividing  the  number  into  periods  of  two  figures  each,          12'96(36 
•we  ascertain  that  there  will  be  two  figures  in  the  root.  9 

The  greatest  square  in  the  left-hand  or  ten's  period  be-     66)  396 
ing  9  hundreds  its  corresponding  root  is  3  tens,  which  395 

we  write  on  the  right  and  subtract  its  square  (9)  from 
the  left-hand  period  (12.)\  The  remainder  with  the  next  period  an- 
nexed is  396.  This  must  contain  the  root  (30)  already  found  multi- 
plied twice,  \or  30X2  multiplied  once  into  the  next  figure,  also  the 
square  of  that  figure  ;  therefore  396-|-60  allowing  for  the  addition  of 
the  remaining  square,  equals  the  next  figure  (6)  of  the  root.  This 
figure  we  also  add  to  the  divisor,  that  it  may  be  squared  as  well  as 
multiplied  into  60.  After  multiplying,  there  being  no  remainder,  36 
is  found  to  be  exactly  the  square  root  of  1296. 

Regarding  1296  as  the  contents  of  a  square  surface,  36  is  the  length 
of  each  side. 


36        6X6  =  * 

I2=    36 

38  3X6  = 
2l6or  6X3  = 
108  3X3  =  3 

18 
18 
*=  9 

1296 

12'96 

EVOLUTION. 


257 


KULE  FOE  EXTRACTING  THE  SQUARE  ROOT.  —  Separate  the 
numbers  into  periods  of  two  figures  each;  beginning  at  the 
right  of  whole  numbers,  and  the  left  of  decimals. 

Find  the  greatest  square  in  the  left-hand  period,  and  placing 
its  root  at  the  right,  subtract  it  from  that  period,  and  bring 
down  the  next. 

Double  the  root  already  found,  and  regarding  its  local  value, 
find  how  many  times  it  is  contained  in  the  remainder  of  the 
given  square.  Annex  the  quotient  to  the  root  and  to  the  di- 
visor, then  multiply  the  the  divisor  by  it,  subtract  the  product 
and  bring  down  the  next  period. 

Proceed  thus  tul  the  figures  in  the  root  are  equal  in  number 
to  the  periods.  If  there  is  still  a  remainder,  periods  of  deci- 
mal ciphers  may  be  supplied. 

To  find  the  root  of  a  common  fraction,  reduce  it  to  its  lowest  terms 
and  extract  the  root  of  the  numerator  and  denominator  separately  if 
they  are  complete  squares  ;  otherwise  reduce  the  fraction  to  a  deci- 
mal and  extract  its  root. 

MENTAJJ  EXERCISES. 

What  is  the  square  root  of  4,  16,  9,  25,  49,  36,  6±,  100,  81, 
121,  144,  |,  &,  &,  rife.  •»>  T¥?>  •»,  -25  ? 


EXAMPLES. 

What  is  the  square  root  of 


(2  ) 

9801 

(8  )  . 

12321 

(3.).. 

4489 

(9.).. 

...8.1225 

6561 

(10  )  . 

.  .  .  40401 

(5.).. 

8649 

(11  ).. 

...  .0695 

(6.;.. 

..7225 

(12.).. 

.  56644 

(13.) 531441 

(14.) 5499025 

(15.) 36372961 


(16.), 
(17.), 
(18.) 


>m 


.0045369 


Art,  194,— Applications  of  the  Square  Root. 

The  areas  of  circles,  squares,  and  all  similar  figures,  are  to  each 
other  as  the  squares  of  their  corresponding  dimensions.  A  circle 
whose  diameter  is  4  feet,  is  to  a  circle  whose  diameter  is  2  feet,  as  4* 
to  22,  or  16  to  4,  four  times  greater. 


258  EVOLUTION. 

A  Triangle  is  a  figure  bounded  by  three  straight  lines, 
The  difference  in  the  direction  of  two  lines  which  meet, 
is  called  an  angle ;  and  if  one  line  is  perpendicular  to  the 
other,  the  angle  is  a  right  angle. 

A  Bight-angled  Triangle  is  a  triangle  that 
has  a  right  angle  ;  as  A  B  C. 

The  Ilypotlieniise  of  a  right-angled   tri- 
angle is  the  side  opposite  the  right  angle  ;  as     if 
A  C.    The  Base  is  the  horizontal  line  ;  as  B  C.     The  other 
is  the  Perpendicular  ;  as  B  A. 

It  is  demonstrated  in  geometry, 
that  the  square  of  the  hypothenuse 
is  equal  to  the  sum  of  the  squares 
of  the  other  two  sides. 

This  may  be  illustrated  by  the  opposite 
figure.  The  small  squares  are  all  square 
inches  or  feet,  &c. ;  and  those  on  the  hy- 
pothenuse are  equal  to  those  on  the  other 
two  sides :  25=16+9. 

BULB  FOR  FINDING  THE  HYPOTHENUSE. — Add  the  squares 
of  the  base  and  perpendicular  and  extract  the  square  root  of 
the  sum. 

BULE  FOR  FINDING  EITHER  THE  BASE  OR  PERPENDICULAR.— 
Subtract  from  the  square  of  the  hypothenuse  the  square  of  the 
other  given  side,  and  extract  the  square  root  of  the  remainder. 

EXAMPLES. 

19.  How  long  must  a  ladder  be,  to  reach  to  the  top  of  a 
house  40  feet  high,  when  its  foot  is  placed  30  feet  from  the 
house  ? 

20.  A  ladder  50  feet  long,  and  having  its  foot  30  feet  from  a 
house,  just  reaches  the  top  ;  what  is  the  height  of  the  house  ? 

21.  A  ladder  50  feet  long  just  reaches  the  top  of  a  house  40 
feet  high  ;  how  far  is  its  foot  from  the  house  ? 


SQUARE   KOOT.  259 

22.  A  room  is  16  feet  long  and  12  feet  wide  ;  what  is  the 
diagonal  distance  between  its  opposite  corners  ? 

23.  Two  ships,  one  sailing  directly  north  and  the  other  di- 
rectly west,  are  100  miles  apart,  both  having  sailed  the  same 
distance  from  the  same  place  ;  how  far  have  they  sailed  ? 

Art.  195. — The  side  of  a  square  equal  to  any  given  surface^ 
is  found  by  extracting  the  square  root  of  the  surface. 

Kectangles,  whose  length  is  a  certain  number  of  times 
greater  than  their  breadth,  may  be  divided  into  that 
number  of  squares,  with  sides  equal  to  the  breadth  of 
the  rectangles. 

24.  It  takes  25  square  yards  of  carpeting  to  cover  a  room  ; 
how  many  feet  square  is  it  ? 

25.  A  rectangular  field  containing  20  acres  is  twice  as  long 
as  it  is  wide  ;  what  is  its  length  and  breadth  ? 

Art.  196. — The  areas  of  similar  figures  are  in  proportion 
to  the  squares  of  their  similar  sides  or  dimensions. 

26.  A  man  having  a  grass  plot  16  feet  square,  wishes  to  make 
it  4  times  larger  ;  how  many  feet  long  must  each  side  be  ? 

27.  A  man  having  water  conducted  from  a  spring  to  his 
house  by  a  lead  pipe  )^-inch  in  diameter  wishes  to  increase  the 
quantity  four  times  ;  how  large  a  pipe  must  he  use  ? 

The  square  root  of  the  product  of  two  numbers  is  a 
mean  proportional  between  them. 
What  is  the  mean  proportional  between 


(28.)  9  and  25 
(29.)  4  and  16 
(30.)  Sand  18 


(31.)  3  and  27 
(32.)  1  and  49 
(33.)  5  and  20 


(34.)  2  and  8 
(35.)  4  and  9 
(36.)  5  and  125 


Promiscuous  Examples  in  Square  Root, 

37.  A  man  about  to  build  a  house  32  feet  square,  wishes  to 
have  the  peak  12  feet  higher  than  the  plate  beams  ;  how  long 
must  the  rafters  be  ? 

38.  A  man  having  1764  peach  trees,  wishes  to  set  them  out 


260  EVOLUTION. 

in  a  square  field,  so  that  it  shall  be  exactly  filled  with  the 
trees  ;  how  many  must  be  in  row  each  way  ? 

39.  A  man  having  1152  apple  trees,  wishes  to  set  them  in 
rows  twice  as  long  one  way  as  the  other ;  how  many  must  be  in 
a  row  each  way  ? 

40.  A  farmer  having  a  ditch  3  fee.t  deep  and  2  feet  wide* 
wishes  to  make  it  twice  as  large  in  the  same  proportion  ;  how 
deep  and  wide  must  it  be  ? 


Cube  Root. 

Art.  197. — The  cube  root  of  a  number,  is  another  num- 
ber, which  multiplied  by  itself  twice,  will  produce  the 
given  number,  or  cube. 

The  cube  of  1  is  1  therefore  the        l/\  is  1 

"  "  2  "  8  "  4/8  "  2 

«  "  3  "  27  "  4/27  "  3 

«  «  4  «  64  "  4/64  "  4 

"  "  5  "  125  "  4/125  "  5 

"  "  6  "  216  "  s/216  "  6 

«  «  7  «  343  «  a/343  «  7 

"  "    8  "    512  "  4/512  "    8 

"  "    9  "    729  "  4/729  "    9 

"  "  10  "  1000  "  v'lOOO  «« 10 

To  find  the  cube  root  of  any  large  number,  we  ob- 
serve— 

First — That  in  cubing  a  number,  each  figure  is  cubed,  and  multi- 
plied three  times  into  the  squares  of  each  of  the  others,  also  its  square 
is  multiplied  three  times  into  each  of  them. 

Secondly — That  the  cube  of  any  number  contains  as  many  periods 
of  three  figures  each,  beginning  at  the  right,  as  there  are  figures  in 
the  root,  except  the  left-hand  period  may  have  only  one  or  two 
figures. 

Thirdly— That  the  cube  of  a  unit's  figure  is  contained  in  the  cor- 
responding unit's  period  ;  of  a  ten's  figure  in  the  ten's  period,  &c. ; 
also,  that  the  cube  of  the  left  hand  figure  is  the  greatest  cube  in  its 
corresponding  period. 


CUBE  ROOT. 


261 


The  cube  of  any  number  from 

1  to      9  is  from  1  to  '729  and  has  1  period. 

10  to    99  is  from       1000  to        '970  299  and  has  2  periods. 

100  to  999  is  from  100000  to  997'002  999  and  has  3  periods. 

Therefore  the  cube  root  of  any  number  from 

1  to  '729  is  from      1  to      9,  1  figure. 

1000  to         970'299  is  from    10  to    99,  2  figures. 
1000000  to  997'002'999  is  from  100  to  999,  3  figures. 
Or  we  may  consider  any  cubic  number  as  representing 
the  contents  of  a  cube,  in  cubic  inches,  feet  &c. 
The  cube  of  36  is— 


36        (36)2 

36 

"210  6X6 
108  30X6 
19QR  nr  6X30 

36      30X30 


7776 


46656 


6X6  X6  =  63           =  216 

30X6  X6  =  30    X62=  1080 

6X30X6  =  30    X62=  1080 

30X30X6  =(30) 2        =  5400 

6X6  X30=  30    X62=  1080 

30X6  X30=(30  2X6  =  5400 

2x6  =  5400 

2        ==  27000 
46656 


30X30X30=(30 


(36) 


To  find  the  cube  root  of  46656,  we 
reverse  this  process. 

Dividing  the  number  into  periods  of  302X3      =2700 

three  figures,  except  the  left-hand  period,  30  X6X3=  54°  46'656(36 

we  ascertain  that  the  root  has  two  figures.  62  = 36  27 

The  greatest  cube  in  the  left-hand  period  3275  )r9656 

being  27  thousands,  its    corresponding  19656 
root  is  3  tens,  which  we  write  on  the 


262  EVOLUTION. 

right,  and  subtract  its  cube  from  the  period.  The  remainder  with  the 
next  period  annexed  is  19656.  This  must  contain  three  times  the 
square  of  the  root  (30,  already  found)  multiplied  by  the  next  root  figure 
(to  be  found, )  three  times  the  root  (found, )  multiplied  by  the  square 
of  that  figure,  and  its  cube  ;  therefore  19656-^-2700,  allowing  for  an  in- 
crease of  the  divisor,  equals  (6)  the  next  root  figure.  Three  times 
this  figure  multiplied  by  the  former  part  of  the  root,  we  add  to  the 
divisor,  also  its  square  that  in  multiplying,  its  square  thus  multiplied 
and  its  cube  may  be  included  in  the  product.  The  divisor,  thus  in- 
creased, is  (  2700  -f  540  -|-  36  )  3276  which  multiplied  by  (6)  the 
last  root  figure  equals  19656  the  lust  remainder.  Therefore  the  cube 
root  of  46656  is  exactly  36. 

EULE  FOR  EXTRACTING  THE  CUBE  BOOT. — Separate  the 
given  number  into  periods  of  three  figures  each,  beginning 
at  the  right  of  whole  numbers,  and  the  left  of  decimals. 

Find  the  greatest  cube  in  the  left-hand  period,  and  placing 
its  root  at  the  right,  subtract  it  from  that  period,  and  bring 
down  the  next  for  a  dividend. 

Write  three  times  the  square  of  the  root  already  found,  with 
a  cipher  annexed,  for  a  trial  divisor,  and,  allowing  for  its 
increase,  find  how  many  times  it  is  contained  in  the  dividend, 
and  annex  the  quotient  to  the  root.  Add  three  times  the  pro- 
duct of  the  last  root  figure  with  a  cipher  annexed,  and  the 
former  part  of  the  root,  to  the  trial  divisor,  also  the  square  of 
the  last  root  figure.  Multiply  the  completed  divisor  by  the  last 
figure  in  the  root,  subtract  the  product  from  the  dividend  and 
bring  down  the  next  period. 

Proceed  thus,  till  the  figures  in  the  root  are  equal  in  number 
to  the  periods. 

Treat  fractions  as  in  square  root. 

EXAMPLES. 

What  is  the  cube  root  of 

Ex.  (1.)  205379  (5.)  80.763 

(2.)  614125  (6.)  29.503629 

(3.)  41421736  (7.)  146363.183 

(4.)  162771336  (8.)  122615327232 

(9.)   t&       (10.)   ^       (11.)  y*W       (12-)   5JM       (13.)   §» 


CUBE   EOOT.  263 

Artt  198. — The  contents  of  all  similar  solids  are  to  each 
other  as  the  cubes  of  their  corresponding  dimensions. 

14.  What  is  the  side  of  a  cube  containing  1728  cubic  feet  ? 

15.  What  is  the  side  of  a  cube  equal  to  a  block  36  in.  long, 
8  in.  wide,  6  in.  high  ? 

16.  If  a  ball  3  inches  in  diameter  weighs  24  Ibs.,  what  will 
be  weight  of  a  similar  ball  6  inches  in  diameter  ? 

17.  A  man  wishes  to  have  a  cubical  cistern  made,  which  will 
hold  25hogsh'ds;  what  must  be  its  depth,  &c.,  allowing  231 
cubic  inches  to  a  gallon  ? 

18.  A  stack  of  hay  16  feet  high,  is  worth  $50  ;  what  is  the 
value  of  a  stack  of  similar  shape,  20  feet  high  ? 

19.  A  farmer  wishes  to  have  a  cubical  box  made  that  will 
hold  100  bushels  of  grain;  allowing  2150.4  cu.  in.  to  a  bushel, 
what  must  be  its  depth,  &c.  ? 

20.  A  cistern  15  feet  deep,  holds  1410.048  gals.;  how  deep 
must  a  cistern  of  similar  shape  be  to  hold  half  the  quantity  ? 


Promiscuous  Examples  in  Square  and  Cube  Root, 

1.  The  pole  of  a  circular  tent  96  feet  in  diameter  is  36  feet 
high;  what  must  be  the  length  of  a  rope  that  will  reach  from 
the  top  of  the  pole  to  the  circumference  of  the  tent  ? 

2.  A  speculator  has  bought  1000  acres  of  western  land,  to  be 
laid  out  in  a  square  ;  what  must  be  the  length  of  its  sides  ? 

3.  A  man  wishes  to  have  a  cubical  ice-house  under  ground, 
that  will  contain  4096  solid  feet  of  ice  ;  what  must  be  its  inside 
dimensions  ? 

4.  A  gentleman  wishes  to  have  a  new  house  built  with  a 
foundation  twice  as  large,  but  in  the  same  proportions,  as  the 
old  one,  which  is  40  feet  long  and  30  feet  wide ;  what  must  be 
its  length  and  width  ? 

5.  What  is  the  mean  proportional  between  45  and  96  ? 

6.  If  a  cylindrical  cistern  6  feet  in  diameter  will  hold  30  hogs- 


264  EVOLUTION. 

heads  of  water,  how  many  gallons  will  a  similar  cistern  hold, 
whose  diameter  is  12  feet  ? 

7.  A  general  has  an  army  of  5184  men  ;  how  many  must  he 
place  in  rank  and  file  to  form  them  into  a  square  ? 

8.  If  a  pipe  2>2'  in.  diameter  discharges  10  hogsheads  of 
water  in  an  hour,  what  must  be  the  diameter  of  another  pipe, 
to  discharge  40  hhds.  in  the  same  time  ? 

9.  What  is  the  side  of  a  cubical  box  that  will  hold  just  one 
bushel  (1250. 4  cu.  in.)? 

10.  A  certain  square,  containing  20736  sq.  feet,  is  paved 
with  stones  a  foot  square ;  how  many  are  there  in  a  row  across 
one  of  its  sides  ? 

11.  If  a  piece  of  silver  3  inches  in  length,  is  worth  $150 ; 
what  is  a  similar  piece  worth  which  is  6  inches  long  ? 

12.  The  area  of  a  circle  80  feet  in  diameter,  is  5026.56  sq. 
feet ;  what  is  the  area  of  a  circle  whose  diameter  is  60  feet  ? 

13.  A  sphere  30  in.  in  diameter  contains  8  cu.  ft.  313.2  cu. 
in. ;  what  is  the  solidity  of  a  sphere  50  feet  in  diameter  ? 


PROGRESSION. 

Art.  199. — Progression  is  a  series  of  numbers  increas- 
ing or  decreasing  uniformly.  There  are  two  kinds,  Arith- 
metical and  Geometrical. 

The  Terms  are  the  numbers  forming  th«  series ;  the 
first  and  last  of  which  are  called  the  Extremes^  the  others, 
the  Means. 

When  the  terms  increase  the  series  is  Ascending  ;  when 
they  decrease,  Descending. 

Arithmetical  Progression. 

Art.  200.— Arithmetical  Progression  is  a  series  of 
numbers  increasing  or  decreasing  by  a  common  difference  / 
as,  2,  4,  6,  8,  10,  12,  9,  6,  3. 


ARITHMETICAL    PEOGBESSION.  265 

In  arithmetical  progression  it  is  required  to  find  the 
first  term  (a),  the  last  term  (I),  the  common  difference 
(d),  the  number  of  terms  (w),  or  the  sum  of  the  series  (s), 
of  which  three  must  be  given. 

In  the  series 

ascending,        3,     5,     7,   9,   11,  13,  15, 

descending,  15,  13,  11,  9,  7,  5,  3,  if  the  common  difference  (2)  be 
multiplied  by  the  number  representing  any  term,  less  1,  and  added  to  or 
subtracted  from  the  first  term,  it  will  give  the  term  sought. 

Again,  half  the  sum  of  the  first  and  last  terms  is  the  average  of  all 
the  terms,  which,  multiplied  by  the  number  of  terms,  equals  the  series. 
Hence  the  following  rules  : — 

Art,  201. — To  find  the  last  term,  the  first,  the  common 
difference,  and  the  number  of  terms  being  given. 

Multiply  the  common  difference  by  the  number  of  terms 
less  1,  and  (in  an  ascending  series)  add  the  product  to  the 
first  term,  or  subtract  it  (in  a  descending  series). 
l  =  a±dx(n-l) 

Art,  202, — To  find  the  common  difference,  the  extremes 
and  number  of  terms  being  given. 

Divide  the  difference  of  the  extremes  by  the  number  of 
terms  less  1. 

I  —  a  or  a  —  I 
n-l 

Art,  203, — To  find  the  number  of  terms,  the  extremes 
and  common  difference  being  given. 

Divide  the  difference  of  the  extremes  by  the  common 
difference,  and  add  one  to  the  quotient. 

I  —  a  or  a  —  I 

•--   -3 —  +  1 

Art,  204. — To  find  the  sum  of  the  series,  the  extremes 
and  number  of  terms  being  given. 

Multiply  half  the  sum  of  the  extremes  by  the  number  of 

terms. 

12 


266  GEOMETRICAL   PROGRESSION. 

a  +  I 
s  =  — —  x  n 

Many  more  cases  might  be  added. 

EXAMPLES. — 1.  What  is  the  last  term  of  a  series  whose  first 
term  is  2,  common  difference  3,  and  the  number  of  terms  25  ? 

2.  What  is  the  common  difference  in  a  series  whose  first  term  is 
2,  last  term  200,  number  of  terms  10? 

3.  What  is  the  number  of  terms  of  a  series,  whose  extremes 
are  2  and  32 ;  common  .difference  3  ? 

4.  What  is  the  sum  of  a  series  whose  extremes  are  3  and  23 ; 
the  number  of  terms  11  ? 

5.  I  agree  to  give  a  man  for  work  3  cents  the  first  hour,  V  cts. 
for  the  second,  11  cts.  for  the  third,  &c.,  for  10  hours;  what 
will  he  receive  for  the  last  hour  ? 

6.  A  man  had  nine  children  at  equal  intervals,  the  oldest  35, 
and  the  youngest  3  years  old ;  what  was  the  difference  in  their 


7.  A  man  traveled  on  foot  the  first  day  5  miles,  and  the  last  45, 
increasing  his  journey  each  day  4  miles  ;  how  many  days  did  he 
travel? 

8.  How  many  times  does  a  clock  strike  in  twelve  hours  ? 

Geometrical  Progression. 

Art.  205, — Geometrical  Progression  is  a  series  of  num- 
bers increasing  by  a  common  multiplier,  or  decreasing  by 
a  common  divisor. 

In  geometrical  progression  it  is  required  to  find  the  first 
term  (a),  the  last  term  (£),  the  ratio  (r),  the  number  of 
terms  (n),  and  the  sum  of  the  series  (s). 

In  the  series 

ascending,       3,     6,  12,  24,  48,  96,  or 

descending,  96,  48,  24,  12,  6,  3,  if  the  ratio  (2  or  £)  be  raised  to  a 
power  one  less  than  the  number  of  any  term,  and  multiplied  by  the  first 
term,  the  product  will  be  the  other  term. 

Again,  if  we  multiply  the  ascending  series  by  the  ratio,  we  shall  have 
another  series,  6,  12,  24,  48,  96,  192,  twice  as  great  as  the  other, 

3,  6,  12,  24,  48,  96,        .     The  difference  between  them 


MENSURATION.  267 

(192  —  3,  the  other  terms  being  alike  in  both  series),  189,  is  therefore 
the  sum  of  the  given  series.     Hence  the  following  rules: — 

Art.  206. — To  find  the  last  term,  the  first  term,  the  ratio, 
and  number  of  terms  being  given  : — 

Multiply  the  first  term  by  that  power  of  the  ratio  whose 
index  is  one  less  than  the  number  of  terms. 

l=a  x  rn  - 1 

Art,  207. — To  find  the  sum  of  the  series,  the  extremes 
and  number  of  terms  being  given  : — 

Multiply  the  last  term  by  the  ratio,  and  divide  the  differ- 
ence between  the  product  and  first  term  by  the  difference 
between  the  ratio  and  1. 

The  last  term  is  frequently  to  be  first  found. 

(iXr)  —  a 

~  r  —  1  or  1  —  r 
Many  more  cases  might  be  added. 

EXAMPLES.— 1.  What  is  the  12th  term  of  a  series  whose  first 
term  is  3,  and  the  ratio  2  ? 

2.  "What  is  the  sum  of  a  series  whose  first  term  is  2  ;  ratio  3  ; 
and  number  of  terms  10? 

3.  A  man  gave  his  son  $100  when  he  was  20  years  old,  and 
promised  to  give  him  $200  the  next  year,  and  to  double  the 
sum  every  year  for  ten  years ;  how  much  did  he  give  his  son 
when  he  was  30  years  old  ?  and  how  much  in  the  ten  years? 

MENSURATION. 

Art.  208. — Mensuration  is  finding  the  length  of  lines, 
and  the  contents  of  surfaces  and  solids. 

The  rules  for  Mensuration  being  derived  from  Geometry,  can  not  be 
explained  by  Arithmetic. 

Mensuration  has  already  been  applied  to  squares,  cubes,  &c.  It  may 
also  be  applied  to  other  surfaces  and  solids,  as 

Parallelograms* — A  parallelogram  is   any   four-sided 


268  MEN8UKATION. 

figure  whose  opposite  sides  are  equal  and  parallel.     They 
include   squares   and  rect- 
angles. 

The  Base  is  the  line  (AB) 
on  which  the  figure  appears 
to  stand.     The  Altitude  is  the  length  of  a  perpendicular 
line  (CD)  from  the  base  to  the  opposite  side. 

Art,  209. — To  find  the  area  of  a  parallelogram. 

RULE. — Multiply  the  base  by  the  altitude. 

Triangles.— (Art.  194.)  The  altitude 
of  a  triangle  is  the  length  of  the  per- 
pendicular drawn  to  the  base,  or  the 
base  produced  from  the  opposite  angle.  B 

Art,  210. — To  find  the  area  of  a  triangle. 

RULE. — Multiply  the  base  by  half  the  altitude. 

Or,  when  the  three  sides  are  given,  from  half  the  sum 
subtract  each  side  separately ;  multiply  together  the  re- 
mainders and  half  the  sum,  and  extract  the  square  root  of 
the  product. 

Circles,— (See  Art.  51.) 

To  find  the  circumference. 

RULE. — Multiply  the  diameter  by  3.14159. 

Art,  211,— To  find  the  diameter. 

RULE. — Multiply  the  circumference  by  .3183. 

Art.  212.— To  find  the  area. 

RULE. — Multiply  the  square  of  the  diameter  by  .7854 ; 
or  the  square  of  the  circumference  by  .07954. 

Cylinders. — A  cylinder  is  a  body  whose  diame- 
ter is  uniform,  and  whose  ends  are  parallel  circles. 

Art,  213. — To  find  the  surface  of  a  cylinder. 

RULE. — Multiply  the  circumference  by  its  length 
or  height,  and  add  to  the  product  the  area  of  the 
two  ends.  (Art.  212.) 


MENSUKATTON.  269 

Art.  214,— To  find  the  solidity. 

RULE. — Multiply  the  area  of  either  end  by  the  length  or  height. 
The  areas  and  solidity  of  prisms  may  be 
found  by  the  same  rules. 

A  Prism  is  a  solid  whose  sides  are  rect- 
angles, and  whose  ends  are  similar  and 
equal. 

A  Pyramid  is  a  solid  having  triangular 
sides  meeting  at  a  point  at  its  top  called 
its  vertex. 

A  Cone  is  a  solid  having  a  circular  base 
and  tapering  to  a  point  at  its  top  called 
its  vertex. 

A  Frustum  of  a  pyramid  or  cone  is 
the  part  which  remains  after  the  top  is 
cut  off  by  a  plane  parallel  to  the  base. 

Art, 2 15,— To  find  the  lateral  surface  of  a  pyramid  or  cone : 

KULE. — Multiply  the  perimeter  or  circumference  of  the 
base  by  one-half  its  slant  height. 

To  find  the  solidity  of  a  pyramid  or  cone  : 

RULE. — Multiply  the  area  of  its  base  by  one-third  its  height. 

Art*  216, — To  find  the  surface  of  a  frustum  : 

RULE. — Add  the  perimeters  or  circumference  of  the  two 
ends  together,  and  multiply  the  sum  by  one-half  the  slant 
height ;  to  the  product  add  the  areas  of  the  ends. 

To  find  the  solidity  of  a  frustum  : 

RULE. — Add  the  areas  of  the  ends  to  the  square  root  of  their 
product,  and  multiply  the  sum  by  one-third  the  height. 

Spheres, — A  sphere  is  a  body  every  part  of 
whose  surface  is  equally  distant  from  the 
center. 

Art,  217, — To  find  the  surface  of  a  sphere  : 

RULE. — Multiply  the  square  of  the  diameter  by  3.14159. 


270  MENSURATION. 

Art.  218. — To  find  the  solidity  of  a  sphere. 

RULE. — Multiply  the  cube  of  the  diameter  by  .5236. 

Art,  219, — To  find  the  curved  surface  of  a  cone. 
RULE. — Multiply  the  circumference  of  the  base  by  half 
the  slant  height. 

The  slant  height  differs  from  the  perpendicular  height,  as  the  hypot- 
enuse of  a  right-angled  triangle  from  its  perpendicular. 

Art.  220« — To  find  the  contents  of  a  cask  in  gallons. 

RULE. — Add  two-thirds  the  difference  between  the  head 
and  bung  diameters  to  the  head  diameter  /  or,  .6  if  the 
staves  are  little  curved,  to  find  the  mean  diameter  •  then 
multiply  the  product  of  the  square  of  the  mean  diameter 
into  the  length  by  .0034. 

EXAMPLES. 

1.  One  side  of  a  triangular  field  is  16  rods  long,  and  a  straight 
line  at  right  angles  with  that  side  from  the  opposite  corner  is  20 
rods  ;  how  many  acres  are  there  in  the  field? 

2.  The  sides  of  a  triangular  field  are  respectively  20,  24,  and 
32  rods  long ;  what  are  the  contents  of  the  field  ? 

3.  In  laying  out  a  grass-plot,  I  fastened  a  rope,  36  feet  long,  at 
one  end,  and  with  the  other  end,  extended  the  whole  length  of 
the  rope,  I  described  a  circle ;    what  are  the  contents  of  the 
plot? 

4.  How  many  square  feet  of  sheet-iron  will  it  take  to  make  a 
stove-pipe  18  feet  long  and  6  inches  in  diameter  ? 

5.  How  many  square  feet  of  tin  will  cover  the  ball  of  a  spire 
3  feet  in  diameter? 

6.  How  many  gallons   of  oil  will  a  cylindrical  tank  hold,  that 
is  V£  feet  deep,  and  4  feet  in  diameter,  allowing  231  cubic  inches 
to  a  gallon  ? 

7.  How  many  cubic  feet  of  gas  will  fill  a  balloon  25  feet  in 
diameter  ? 

8.  How  many  acres  are  there  in  a  field  which  can  be  divided 


MENSURATION.  271 

into  two  triangles,  one  having  a  side  32  rods  long,  and  distant  in 
&  straight  line,  at  right  angles  to  it,  24  rods  from  the  opposite 
corner;  the  other  having  its  sides  respectively,  36,  28,  and  20 
rods  long? 

9.  The  water-wheel  of  a  mill  is  24  feet  in  diameter ;  what  is 
its  circumference  ? 

10.  A  circular  pond  is  300  feet  in  circumference ;  how  much 
land  does  it  cover  ? 

11.  The  diameter  of  the  earth  is  8,000  miles  (nearly) ;  suppos- 
ing it  to  be  a  perfect  sphere,  how  many  square  miles  are  there  on 
its  surface  ? 

12.  How  many  sheets  of  tin,  18  inches  long  and  1  foot  wide, 
will  cover  a  conical  spire,  40  feet  high  and  18  feet  in  diameter  at 
its  base,  allowing  £  for  waste  in  cutting  ? 

13.  How  many  cubic  feet  in  a  stick  of  timber.  40  feet  long,  3 
feet  in  diameter  at  the  larger  end,  and  2  feet  6  inches  at  the 
smaller  ? 

14.  How  many  gallons  in  a  cask  whose  bung  diameter  is  18 
inches,  head  diameter  12  inches,  and  length  24  inches  ? 

15.  How  many  square  feet  of  tin  will  be  required  to  make  an 
oil-can,  in  the  form  of  a  frustum  of  a  cone,  12  inches  in  diameter 
at  the  bottom,  and  4  inches  at  the  top,  the  slant  height  being  2£ 
feet,  allowing  ^  for  waste  in  cutting  ? 


PROMISCUOUS  EXAMPLES. 

UNITED   STATES   MONEY  AND  COMPOUND  NUMBERS. 

1.  A  lady  purchased  a  shawl  for  $16,  2  pairs  of  gloves  at 
$1.37^  a  pair,  and  14  yds,  of  silk  ;  she  gave  the  merchant  three 
$20  bills  and  received  back  $16.05  ;  what  was  the  price  of  the 
silk  per  yard  ? 

2.  A  silversmith  had  10  Ibs.  3^  oz.  of  silver,  and  made  it  into 
spoons  ;  these  he  sold  for  $!>£  apiece,  receiving  for  them  all 


I 


272  PEOMISCUOUS  EXAMPLES. 

$42% ;  how  many  spoons  did  he  make,  and  what  was  the 
weight  of  each  ? 

3.  Beduce  3  m.   4  fur.  21  rods,  3  yds.  6  in.  to  inches  and 
prove  the  operation. 

4.  How  many  shingles  will  it  take  to  cover   the  roof  of  a 
house  42  feet  long,  and  30  feet  wide,  whose  rafters  are  18  feet 
long,  allowing  each  shingle  to  be  6  inches  wide  and  to  lie  6 
inches  to  the  weather  ? 

5.  What  will  it  cost  to  build  a  brick  house  36  feet  long,  32 
feet  wide,  and  24  feet,  high  (on  an  average,)  the  walls  1  foot 
thick,  each  brick  being  8  inches  long,  4  inches  wide  and  2 
inches  thick,  at  $6^  per  1000  bricks  ? 

6.  A  man  was  born  at  8)4  o'clock  A.  M.,  Sept.  5,  1835,  and 
died  at  6  o'clock  P.  M.,  April  21,  1863  ;  what  was  his  age  ? 

7.  The  longitude  of  New  York  is  74°  1',  and  that  of  Cincin- 
nati, 84°  24' ;  what  is  the  difference  of  time  ? 

8.  The  difference  of  time  between  Washington  and  St. 
Petersburg  is  7  hours  9  minutes  ;   what  is  the  difference  of 
longitude  ? 

9.  A  farmer  exchanged  3  pairs  of  oxen  at  $112  a  pair,  and  7 
cows  at  $42  each,  for  126  sheep ;  what  was  the  price  of  the 
sheep  per  head  ? 

10.  Bought  of  J.  Ayers  7  yds  of  cloth  at  $6  a  yard,  3  bbls. 
of  flour  at  $9  barrel,  and  a  cheese  for  $3  ;  he  agreed  to  take 
in  payment  6  cords  of  wood  at  $7  a  cord,  and  18  bushels  of 
corn  at  $1;  how  much  cash  must  I  pay  him  to  balance  the 
account  ? 

11.  In  measuring  a  portion  of  a  railroad,  one  man  made  the 
distance  43  m.  7  fur.  31  rds.  1  yd.  l^ft.  ;    and  another  42  m. 
1  fur.  39  rods,  5  ft.  7  in.  ;  what  was  the  difference  in  the  meas- 
urements ? 

12.  In  measuring  a  field,  one  man  made  it  contain  4  acres, 

3  roods,  18  square  rods,  6  feet,  64  square  inches  ;  but  another 

4  acres,  2  roods,  39  square  rods,  8  yards,  100  square  inches ; 
what  was  the  difference  in  the  measurements  ? 

13.  A  certain  railroad  is  56)£  miles  long  ;  in  making  it,  it 


COMPOUND   NUMBERS.  273 

was  divided  equally  between  6  contractors  ;  how  much  of  it 
did  each  make  ? 

14.  The  difference  in  longitude  between  London  and  Boston 
is  71°  8';  what  is  the  difference  of  time  ? 

15.  The  difference  of  time  between  Jerusalem  and  Balti- 
more is  7  h.  24  m.  36  sec. ;  the  long,  of  Baltimore  is  76°  37'  west; 
what  is  the  long,  of  Jerusalem  ? 

16.  How  many  blocks  of  marble  6  inches  square,  will  pave 
two  halls  of  a  hotel,  crossing  each  other  in  the  centre  ;  one 
32  ft.  long,  and  12  ft.  wide,  the  other  64  ft.  long  and  8  feet 
wide  ? 

17.  In  a  town  5  miles  square,  how  many  farms  can  there  be 
containing  150  acres  each  ? 

18.  A  man  has  agreed  to  deliver  48  cords  of  wood  ;  the 
wood  is  4  feet  long,  and  he  makes  the  pile  5  feet  high  ;  how 
long  must  he  make  it  ? 

19.  A  man  has  $1623  which  is  4  times  as  much  as  he  had 
last  year,  wanting  $121  ;  his  brother  had  last  year  3  times  as 
much  as  he  had,  and  $10.50  more,  but  has  since  lost  $1000  ; 
how  much  has  his  brother  left  ? 

20.  A  merchant  having  $1000,  paid  $510  for  dry  goods  and 
the  remainder  for  20  barrels  of  molasses  ;  how  much  was  the 
molasses  per  barrel  ? 

21.  How  many  suits  of  clothes,  each  requiring  4  yds.  2  qrs. , 
can  be  made  from  639  yards  ? 

22.  How  many  bricks  will  be  required  to  pave  a  sidewalk 
80  yds.  long  and  3  yds.  wide,  each  brick  being  8  in.  long,  and 
4  in.  wide  ? 

23.  How  many  yards  of  muslin,  3  qrs.  wide,  will  line  9  yds, 
1  qr.  2  na.  of  merino  cloth,  1  yd.  1  qr.   wide  ? 

24.  How  much  will  it  cost  at  2s.  6d.  a  square  yard  to  plaster 
a  room  24  feet  long,  18  feet  wide,  and  12  feet  high,   there 
being  two  doors  7  ft.  high,  and  3  ft.  6  in.  wide ;  also  two  win- 
dows 5  feet  high,  and  3  feet  4  in.  wide,  and  a  mop-board 
8  inches  wide. 

12* 


274  PROMISCUOUS    EXAMPLES. 

25.  A  man  wishes  to  divide  a  field  of  4  acres  into  7  building 
lots  of  equal  size ;  how  much  will  each  lot  contain  ? 

26.  The  longitude   of  New   York  is   74°    1';  a  sea   captain, 
sailing  thence,  finds  that  his  watch  has  lost  2  hours ;  what  is  his 
longitude? 

27.  The  difference  of  longitude  between  New  York  and  New 
Orleans  is  15°  4'.     What  is  the  difference  of  time? 


28.  A  lady  shopping  in  New  York,  bought  14  yds.  of  silk  at 
15s.  6d.  a  yard;  5  yds.  of  linen  at  7s.  4d.;  2  yds.  of  thread-lace 
at  18s.,  and  a  pair  of  gloves  for  10s.  9d.     She  gave  in  payment  a 
$100  bill ;  how  much  money  should  she  have  been  paid  back  ? 

29.  Purchased  24  A.  1  B.  27  rods  at  $360  an  acre.     I  sold  the 
same  for  building  lots  at  $3.15  a  square  rod ;  what  did  I  gain? 

30.  A  grocer  bought  5  barrels  of  cider  at  $4  a  barrel,  and  after 
making  it  into  vinegar,  sold  it  at  10  cts.  a  quart ;  how  much  did 
he  gain? 

COMMON    AND   DECIMAL   FRACTIONS. 

1.  A  farmer  had  £  of  his  sheep  in  one  pasture,  £  in  another,  £ 
in  another,  and  the  remainder,  46,  in  a  fourth ;  how  many  sheep 
had  he  ? 

2.  A  merchant  bought  6|  cords  of  wood  at  $5£  a  cord  ;  and  paid 
for  it  in  cloth  at  $4£  a  yard ;  how  many  yards  did  it  require  ? 

3.  Bought   640  sheep  at  $2|  a  head,  and  afterward   270  at 
$2f ;  sold  f  of  them  at  $3^;  |  of  the  remainder  were  killed  by 
dogs,  and  what  still  remained  I  sold  for  $3  a  head ;  how  much 
did  I  gain  ? 

4.  What  will  75  yds.  1-J  qrs.  of  silk  cost,  at  £.375  a  yard? 

5.  A  butcher  had  351£  Ibs.  of  beef;  he  sold  f  of  it,  corned  § 
of  the  remainder,  and  used  what  was  then  left  in  his  family ;  what 
was  the  value  of  that  which  his  family  consumed  at  17^  cts.  a 
pound  ? 

6.  What  part  of  £  of  a  solid  yrd.,  is  £  of  a  yard  solid  ? 

7.  How  much  can  a  man  earn  in  f  of  a  year  at  $1£  a  day  ? 


COMMON    AND   DECIMAL    FRACTIONS.  275 

8.  A  piece  of  oil-cloth  3  yds.  square  is  worth  $30.     What  is  3 
square  yards  of  it  worth  ? 

9.  A  farmer  sold  4  fields  containing  respectively,  6^,  8^,  5-|, 
and  7^  acres,  at  $100  an  acre  ;  what  did  he  receive  for  them  ? 

10.  A  hogshead  of  molasses  contained  96  gallons;    -f%  of  it 
lias  been  sold ;  §  of  the  remainder  has  been  used ;  how  much  now 
remains  ? 

11.  Two  men  were  82  miles  apart,  one  of  them  traveled  -fo  of 
this  distance,  and  the  other  ^  of  the  remainder ;  -  how  far  were 
they  then  apart? 

12.  A  laborer  hired  to  a  farmer  for  a  year  for  $313,  but  he 
was  sick  £  of  the  working  time,  and  was  absent  £  of  the  remain- 
ing time;  how  much  wages  was  due  to  him  at  the  end  of  the 
year? 

13.  A  man  had  a  field  36^  rods  long,  and  24f  rods  wide ;  he 
sold  $  of  it  for  $112.40 ;  what  is  §  of  the  remainder  worth  at  the 
same  rate  ? 

14.  A  merchant  sold  $  of  a  hogshead  of  sugar  weighing  879 
Ibs.  for  $96.     What  is  f  of  the  remainder  worth  at  the  same 
rate? 

15.  Bought  a  piece  of  merino  cloth  containing  48f  yards,  and 
having  cut  off  §  of  it,  sold  f  of  the  remainder  at  $!£  a  yard,  and 
what  still  remained  at  87-|  cents  a  yard ;  what  was  the  amount 
sold? 

16.  A  man  having  28£  tons  of  coal,  sold  f  of  it  at  $10f  a  ton, 
and  the  remainder  at  $9f .     What  amount  was  received  for  the 
coal? 

17.  A  man  bought  42^  tons  of  hay,  at  $11^  a  ton,  and  sold 
\  of  it  at  $11|,  and  the  remainder  at  $13£  a  ton ;  how  much  did 
he  gain  ? 

18.  A  grocer  bought  a  cask  of  vinegar  containing  43f  gals,  for 
*15^,  and  has  sold  19  gals.  2  qts.  1^  pts.  at  cost;  for  what  must 
he  sell  the  remainder  in  order  to  gain  £  as  much  as  the  whole 
cost? 

19.  What  cost  .778125  ton  of  buckwheat  flour,  a  2d.  at  pound? 

20.  What  cost  1.8875  acre  of  land  at  $1    a  rod  ? 


276  PROMISCUOUS    EXAMPLES. 

21.  What  cost  112  hhds.  3.35  gals,  of  molasses,  at  £3  8s.  9d.  a 
hogshead  ? 

22.  If  .3125  yard  of  cloth  cost  £$  ;  what  is  the  price  per  yard? 

23.  If  .0625   bbl.   of  flour  cost  $^-;    what  is  the  price  pel- 
barrel  ? 

24.  A  man  having  185f  acres,  sold  ^  of  it,  at  another  time  | 
of  it;  what  is  the  value  of  the  remainder  at  $115f  an  acre? 

25.  A  man  deposited  his  money  in  4  banks,  in  one  f ,  in  another 
J,  in  the  third  £,  and  in  the  fourth  the  remainder,  which  was  $48 
more  than  -jL  of  the  whole ;  how  much  money  did  he  deposit? 

26.  Bought  f  of  a  box  of  starch,  sold  £  of  it  for  $4£f ;  what 
was  the  whole  box  worth  at  the  same  price  ? 

27.  A  man  gave  •£  of  his  money  for  a  horse ;  §  of  the  remainder 
for  a  wagon  ;  |  of  what  then  remained  for  a  saddle;  he  then  had 
$24.25  left;  how  much  had  he  at  first? 

28.  A  man  bequeathed  his  property  to  his  five  children  :  to  the 
first  %  of  it ;  to  the  second  £ ;  to  the  third  £ ;  to  the  fourth  | ; 
and  the  remainder  to  the  fifth,  who  had  $665  less  than  the  fourth  ; 
what  was  the  amount  of  the  property  ? 

29.  A  person  being  asked  the  time  of  day,  said :  The  time  past 
noon  is  f  of  the  time  from  now  to  midnight. 

30.  Two  men,  A  and  B,  were  playing  cards  for  money ;  f  of 
A's  money  was  equal  to  f  of  B's ;  but  B  lost  $42,  and  then  had 
only  -j^  times  f  as  much  as  A  then  had ;  how  much  had  each  ? 


81.  A  man  gave  $1750  to  two  benevolent  societies,  and  gave 
one  $150  more  than  the  ^  of  what  he  gave  the  other;  what  were 
the  amounts  given  ? 

32.  A  man  bequeathed  $5,420  to  his  son  and  daughter,  so  that 
$240  more  than  §  of  what  his  daughter  had  was  equal  to  f  of 
what  his  son  had;  how  much  had  each? 

33.  A  pole  102  feet  high  stands  near  a  house,  and  £  of  the  part 
above  the  house  equals  §  the  other  part ;  how  much  higher  is  the 
pole  than  the  house  ? 

34.  James  said  to  Henry,  2  years  added  to  £  of  my  age  equals 


COMMON    AND    DECIMAL    FRACTIONS.  277 

§  of  yours,  and  the  sum  of  our  ages  is  37  years ;  what  was  the  age 
of  each  ? 

35.  A  person  being  asked  his  age,  replied,  that  f  of  it,  f  of  it, 
£  of  it,  and  7  more,  would  be  twice  his  age ;  what  was  his  age  ? 

36.  The  difference  between  my  neighbor's  property  and  my 
own  is  $1000  ;  f  of  mine  equals  f  of  his ;  but  £  of  his  is  1^  times 
^  of  mine  ;  what  is  the  property  of  each  ? 

37.  I  wish  to  make  3  boxes,  each  5|  feet  long,  3^  feet  wide, 
and  2^  feet  high ;  how  many  square  feet  of  boards  1-|  inches  thick 
will  they  require  ? 

38.  A  man  has  a  garden  10£  rods  long  and  7£  rods  wide ;  what 
will  it  cost  to  dig  a  ditch  around  it  3  feet  wide  and  4£  feet  deep, 
at  3  cents  a  cubic  foot? 

39.  A  man  traveled  28f  miles  the  first  day,  33||  miles  the 
second,  and  29^T  miles  the  third ;  how  far  did  he  travel  in  the 
three  days  ? 

40.  The  distance  from  Boston  to  Worcester  is  40  miles  ;  a  man 
havjng  traveled  -fo  of  the  distance,  and  afterward  -ft  of  it ;  how 
far  was  he  then  from  Boston? 

41.  Bought  23|  bushels  of  corn  at  $lf  a  bushel,  and  sold  f  of 
it  for  $1-|,  and  the  remainder  at  $1-| ;  what  was  the  whole  gain  ? 

42.  Bought  f  of  a  flouring  mill  for  $1,837.50,  and  having  sold 
7  of  my  share,  I  gave  §  of  the  remainder  to  cancel  a  mortgage 
on  it,  and  what  still  remained  I  gave  the  miller  for  ^  a  year's 
wages  ;  what  was  the  amount  of  his  wages  a  year  ? 

43.  Bought  a  horse,  carriage,  and  harness  for  $350.      The  har- 
ness cost  -fy  as  much  as  the   horse,  and  the  horse  f  as  much  as 
the  carriage  ;  what  did  each  cost? 

44.  Bought  I  of  a  ton  of  plaster,  and  sold  -fa  of  it  for  $4.50  ; 
what  was  the  price  per  ton? 

45.  One  of  my  horses  usually  travels  6  miles  in  £  of  an  hour, 
and  the  other  7  miles  in  ^  of  an  hour ;  how  much  longer  will  it 
take  one  to  travel  20  miles  than  the  other  ? 

46.  I  have  a  garden  13f  rods  long,  and  f  as  wide,  surrounded 
by  a  fence  6£  feet  high.     Next  to  the  fence  is  a  border  1  rod 
wide,  for  shrubbery  and  fruit-trees ;  then  a  gravel  walk  8£  feet 


278  PROMISCUOUS    EXAMPLES. 

wide,    the  rest  is   for  cultivation;    how  much   is   there   to   be 
cultivated  ? 

47.  How  many  pieces  of  paper,  9-|  yards  long  and  20  inches 
wide,  will  it  take  to  paper  a  room  22-^  feet  long,  15^  feet  wide, 
and  Hi  feet  high? 

48.  A  man  invested  -£  of  his  property  in  his  business,  £  of  the 
remainder  in  stocks,  ^  of  what  still  remained,  $454,  in  a  farm  ; 
what  part  of  his  property  was  thus  invested,  and  what  was  the 
amount  of  the  whole? 

49.  A  man  invested  \  of  his  property  in  a  farm,  and  £  of  the 
remainder  he  spent  in  building  a  house ;    the  farm  cost  $1000 
more  than  the  house  ;  what  was  the  amount  of  his  property  ? 

50.  After  spending  f  of  my  money,  and  J  of  what  remained, 
I  had  $62.50  left ;  what  sum  had  I  at  first? 

51.  A  cask  was  §  full  of  vinegar  ;  after  drawing  from  it  8  gal- 
lons it  was  \  full ;  how  many  gallons  did  it  hold  ? 

52.  If  f  of  the  time  past  noon  is  f  of  the  time  to  midnight, 
what  time  is  it? 

53.  If  a  man   draw  350  loads  of  bricks,  and  1,500  bricks  at 
each  load ;  how  much  will  he  receive  at  the  rate  of  87^  cents  a 
thousand  ? 

54.  Sold  3000  feet  of  boards,  at  $9.50  a  thousand,  and  700  lath 
at  50  cents  a  hundred,  what  was  the  amount  ? 

55.  Bought  160  sheep  at  $2£  a  head,  and  215  at  $1.87£  a  head, 
sold  §  of  them  at  $2|,  and  the  remainder  at  $2 ;  did  I  gain  or 
lose,  and  how  much  ? 

56.  Bought  364  pounds  of  sugar,  at  16f  cents  a  pound  ;  if  the 
price  per  pound  had  been  3£  cents  less,  how  many  pounds  could 
have  been  bought  for  the  same  money  ? 

57.  My  farm  is  f  meadow,  ^  orchard,  and  the  remainder,  20 
acres,  more  than  £  of  the  whole,  is  timber;  how  large  is  my 
farm? 

58.  The  income  of  a  farm,  consisting  of    125A.  1R.  7$  rods, 
was  £202  11s.  3fd. ;  how  much  was  it  an  acre? 

59.  How  much  wheat  will  32 A.  1R.  10  rods  yield,  at  the  rate 
of  25  bu.  3  pks.  1  qt.,  per  acre  ? 


PERCENTAGE    AND    ITS    APPLICATIONS.  279 

60.  I  have  two  small  farms,  §  of  the  acres  in  one,  added  to  f 
of  the  acres  in  the  other,  make  90  acres  ;  and  §  of  the  first  is  $ 
of  f  of  the  second  ;  how  much  larger  is  the  first  farm  than  the 
second  ? 


PERCENTAGE    AND    ITS    APPLICATIONS. 

1.  Sold  a  quantity  of  grain  for  $251.50,  for  which  I  received 
a  note  dated  Oct.  1,  1862,  payable  in  6  months  ;  what  is  the  value 
of  the  note  Dec.  27  ? 

2.  I  have  forwarded  to  my  agent  in  St.  Louis,  $5000  for  the 
purchase  of  flour ;  after  deducting  ty%  commission,  what  will 
be  the  cost  of  the  flour? 

3.  What  must  be  the  face  of  a  note  payable  in  4  months,  for 
which  I  may  receive  from  a  bank  $600  at  6%  discount  ? 

4.  A  woolen  factory  was  insured  for  $37,500  at  2-|%  ;   after  2 
years  it  was  burnt;  what  was  the  loss  to  the  company  ? 

5.  What  is  the  duty  at  30%  ad  valorem,  on  26  barrels  of  sugar, 
each  weighing  225  pounds;  tare  15%,  and  the  sugar  costing  16 
cents  a  pound  ? 

6.  Bought  25  barrels  of  flour  at  $10  a  barrel,  and  sold  it  im- 
mediately at  $11.96  a  barrel,  on  8  months'  credit,  what  per  cent, 
did  I  gain,  allowing  6%  interest  ? 

7.  Sold  34  tons  of  coal  at  $8  a  ton,  for  which  I  received  a 
note  payable  in  90  days,  and  had  it  discounted  at  a  bank.     I  then 
found  that  I  had  lost  10%  on  the  coal ;  what  did  it  cost? 

8.  If  I  buy  cloth  at  $7.50  a  yard  on  9  months'  credit,  for  what 
must  I  sell  it  for  cash,  to  gain  12%  ? 

9.  Messrs.  Mead,  Gage  &  Storrs,  of  Chicago,  made  a  consign- 
ment of  flour  to  New  York ;  M.  furnished  $1,400 ;  G.  $600,  and 
S.  200  barrels  of  flour.     They  gained  $270,  of  which  S.  had  $90  ; 
at  what  was  his  flour  valued  per  barrel,  what  was  M.'s  and  G.'s 
share  of  fche  profits? 

10.  Three  men  A,  B,  and  0,  hired  a  pasture  for  $72.     A  put 
in  3  horses  for  6  weeks,  B  put  3  pairs  of  oxen  for  5  weeks,  and 
0  put  in  12  cows  for  4  weeks.     It  was  agreed  that  5  cows  should 


280  PROMISCUOUS    EXAMPLES. 

be  considered  equal  to  3  oxen,  and  4  oxen  to  3  horses ;  what 
was  each  one's  share  of  the  expense  ? 

11.  Messrs.  Dudley  &  Swift  contracted   to  build  a  section   of 
a  railroad  for  $26,000  a  mile ;  D.  furnished  60  men,  and  S.  40 
horses  and  carts,  with  boys  to  drive  them.     It  was  agreed  that  3 
men  be  considered  equal  to  2  horses  and  their  drivers.     Swift 
also  was  to  be  allowed  $100  a  mile  for  overseeing  the  work. 
After  completing  5|  miles,  what  was  each  one's  share  ? 

12.  A  merchant  in  Philadelphia  wishes  to  remit  to  .Liverpool 
£1000 ;  what  will  a  bill  of  exchange  for  this  amount  cost  him  at 
9K  %  premium  ? 

18.  When  gold  is  135,  which  is  the  better  investment,  U.  S. 
5-20's  at  103,  or  bank  stock  at  108,  paying  a  semi-annual  divi- 
dend of  4%  ? 

14.  Bought  goods  amounting   to  $256.50,    and   having   kept 
them  6  months,  sold  them  so  that  I  gained  $% ;  for  what  were 
they  sold? 

15.  Borrowed  of  my  neighbor  $450  for  6  months ;    I  after- 
ward lent  him  $300,  long  enough  to  compensate  him  ;  how  long 
did  he  keep  it  ? 

16.  A  certain  town  is  taxed  $3022.75 ;  the  taxable  property 
amounts  to  $146,637.50;  there  are  150  polls,  each  taxed  60  cts. ; 
what  per  cent,  is  the  tax,  and  what  is  a  man's  tax  who  pays  for 
two  polls  and  whose  property  is  valued  at  $1837.50? 

17.  What  is  due  on  the  following  note,  at  *I%  interest,  Jan.  1, 
1868? 

$500.  SING  SING,  K  Y.,  Oct.  10, 1862. 

On  demand,  I  promise  to  pay  S.  Wilbur,  or  order,  five  hundred 
dollars,  value  received,  with  interest.  H.  VAN  WYCK. 

Indorsed, 

Jan.  1,  1863,  $60.  April  1,  1865,  $75. 

June  15,  1864,  $150.  Jan.  1,  1866,  $100. 

18.  Sold  a  lot  of  lumber  for  $500,  and  gained  Vb\% ;  what  did 
it  cost  ? 

19.  Clark  &  Smith  are  partners;  0.  put  in  $2000,  and  they 


PERCENTAGE   AND   ITS    APPLICATIONS.  281 

have  gained  $203,  of  which  Smith's  share  is  $87  ;  what  was  his 
capital  ? 

20.  Lent  $176   for  1  year  6  months;  it  then  amounted  to 
$195.14 ;  what  was  the  per  cent.? 

21.  Sold  several  shares  in  an  Oil  Company  for  $6875,  which 
was  40  per  cent,  less  than  they  cost;  what  did  they  cost ? 

22.  Bought  goods  to  the  amount  of  $1200,  sold  them  for  $1344  ; 
what  was  the  gain  per  cent.  ? 

23.  Paid  $10,000  for  a  cargo  of  cotton,  and  sold  it  for  $15,000, 
but  invested  it  in  stock,  which  I  sold  at  15  per  cent,  less  than  it 
cost ;  what  was  the  net  gain  ? 

24.  Sold  36  bushels  of  corn  for  $29.70,  and  lost  \*l\%\  what 
per  cent,  should  I  have  gained  if  I  had  sold  it  for  $40^  ? 

25.  Bought  30  yards  of  cloth  at  5%  less  than  the  first  cost,  and 
sold  it  at  5%  more  than  the  first  cost ;  I  gained  $15 ;  what  was 
the  first  cost  per  yard  ? 

26.  A  bank  discounted  a  note  payable  in  60  days,  at  6%  dis- 
count ;  and  gave  for  it  $2.52  less  than  the  face  of  the  note  ;  what 
was  the  amount  of  the  note  ? 

27.  Bought  a  house  for  $3000 ;  rented  it  at  $350 ;  paid  $%  for 
insurance,  for  taxes  If/^,  for  repairs  $106 ;  what  per  cent,  did 
the  investment  yield  ? 

28.  A  man  sold  his  farm  for  $2500,  which  was  16f%  less  than 
he  paid  for  it;  .he  then  bought  another,  and  sold  it  for  16%"  more 
than  he  paid  for  it ;  he  thus  gained  as  much  as  he  had  lost ; 
what  did  he  pay  for  each  farm  ? 

29.  A  merchant  bought  120  yards  of  cloth,  at  $4  a  yard,  on  6 
months'  credit,  and  sold  it  immediately  for  $500,  money  being  at 
6%;  what  did  he  gain? 

30.  The  tax  in  a  certain  town  is  \\%  besides  each  poll  $1.     One 
man's  tax  is  $127,  including  2  polls ;  what  is  the  amount  of  his 
property? 

31.  A  collector  received  $36  for  his  services,  at  %\%\  what 
was  the  amount  he  collected  ? 

32.  A  man  bought  a  house,  and  after  spending  10%  of  the 


282  PBOMI6CTJOTTS    EXAMPLES. 

price  in  repairs,  found  that  the  whole  cost  was  $4400  ;  what  was 
the  price  of  the  house  ? 

33.  Bought  a  house  for  $4000  ;  paid  for  repairing  it  $1500,  it 
remained  unoccupied  3  months,  when  I  sold  it  for  $6000,  for 
which  I  received  a  note  payable  in  90  days,  after  one  month  I 
had  the  note  discounted  at  a  bank  at  *1%\  what  per  cent,  did  I 
gain  by  the  purchase  ? 

34.  Lent    my  neighbor  $900  from  Jan.    1  to  Sept.  1 ;    then 
borrowed  of  him  $1150  from  Sept.  1  to  March  1 ;  what  is  the 
balance  of  interest  at  6%,  and  to  whom  due  ? 

35.  What  investment  at  5%  will  yield  a  semi-annual  income  of 
$250  ? 

36.  I  have  bought  a  bill  of  goods  amounting  to  $420,  on  60 
days'  credit.     Is  it  better  for  me  to  pay  cash  at  2^%  discount,  or 
receive  Q%  interest  for  the  money  till  the  time  of  credit  expires? 

37.  In  order  to  pay  the  above  bill  is  it  better  for  me  to  obtain 
the  money  from  a  bank  at  6%  discount,  or  take  the  goods  on 
credit  ? 

38.  Exchanged  100  shares  of  Erie  stock  at  40%  below  par,  for 
stock  in  a  Gold  Mining  Company  at  125 ;  how  many  shares  did 
I  receive  ? 

39.  Gained  $175  by  buying  50  shares  of  bank  stock,  at  5% 
advance,  and  selling  them  again :   for  how  much  a  share  were 
they  sold? 

40.  A  man  has  $4000,  invested  in  U.  S.  5-20's ;  what  income 
will  it  yield  when  gold  is  130  ? 

41.  Bought  1%  bonds  at  103,  amounting  to  $7210;  what  an- 
nual income  will  they  yield  ? 

42.  How  much  must  be  invested  in  6%  bonds  at  90,  to  yield  a 
semi-annual  income  of  $500  ? 

43.  If  I  buy  6%  bonds  at  90,  amounting  to  $6000,  what  %  will 
the  investment  yield  ? 

44.  Which  is  the  better  investment,  D".  S.  7-30's  at  102,  or  6% 
State  bonds  at  98  ? 

45.  What  must  be  the  price  of  gold  that  U.  S.  5%  bonds  at  95 
may  yield  6%  interest  ? 


PERCENTAGE    AND   ITS    APPLICATIONS.  283 

46.  A  bankrupt  settled  with  his  creditors  by  paying  them  70 
cents  on  a  dollar ;  one  o'f  them  received  $507.50  ;  what  was  the 
amount  of  his  claim  ? 

47.  What  will  be  the  premium  at  l^%  for  insuring  a  vessel  to 
cover  both  its  value  $10,000  and  the  premium? 

48.  At  the  age  of  40  a  man  insured  his  life  for  $5000,  at  the 
rate  of  $36  a  1000,  not  to  be  paid  after  20  years.     "What  will  his 
family  gain  or  lose  if  he  die  at  45,  50,  55,  60,  or  afterward,  money 
being  worth  6%  ? 

49.  How  much  must  be  paid  in  currency  for  duties  on  25  bar=" 
rels  of  sugar,  each  weighing  224  Ibs.,  tare  12,%,   and  the  duty 
5  cts.  a  pound  in  gold,  when  gold  is  130? 

50.  A  gentleman  in  St.  Louis  owns  50  shares  of  the  Corn 
Exchange  Bank  in  New  York,  which  has  declared  a  semi-annual 
dividend  of  5%",  a  draft  for  which  he   sells  at  \%  premium  ; 
what  does  he  receive  for  it  ? 

51.  A  merchant  in  New  York  owes  a  debt  in  Liverpool  of 
£250.     When  gold  is  130,  and  exchange  9^,  is  it  better  to  buy  a 
bill  of  exchange,  or  remit  U.  S.  bonds  at  95,  which  can  be  sold 
in  Liverpool  at  60  ? 

52.  A  man  left  $10,000  to  be  divided  between  his  two  sons, 
16  and  18  years  old,  so  that  at  6%  interest  they  should  each 
have  the  same  amount  when  21  years  old  ;  what  did  he  leave  for 
each? 

53.  A  man  owed  $287.70;  he  paid  at  one  time  40%  of  the 
debt ;  at  another  time  25%  of  what  he  then  owed ;  and  after- 
ward 12-^%"   of  what  he  still  owed;     how  much  of  the  debt 
remained  to  be  paid  ? 

54.  A  merchant  having  $5000,  lost  |  of  it  in  speculation,  and 
^  of  the  remainder  in  bad  debts ;  what  per  cent,  of  the  whole 
did  he  lose  ? 

55.  A  man  spent  $487.50  in  traveling,  which  was  15%  of  his 
income  ;  what  was  his  income  ? 

56.  A  man  exchanged  14  shares  of  bank  stock,  at  T%  premium, 
for  25  shares  of  railroad  stock,  at  12^%  discount,  and  agreed  to 
pay  the  difference  in  cash ;  how  much  did  he  pay  ? 


284  PROMISCUOUS   EXAMPLES. 

MISCELLANEOUS  RULES. 

[These  include  various  rules  not  used  in  the  last  promiscuous  examples.] 

1.  I  have  3  rooms,  respectively  12,  16,  and  20  feet  wide,  which 
I  wish  to  cover  with  oil-cloth  that  will  exactly  fit  all  of  them 
without  cutting  off  the  width  ;  how  wide  must  the  oil-cloth  be  ? 

2.  I  have  3  pieces  of  oil-cloth,  respectively  3,  6,  and  9  feet 
wide  ;  what  must  be  the  width  of  a  room  that  either  of  them  will 
exactly  fit  without  cutting  off  the  width  ? 

3.  What  will  it  cost  to  gild  a  ball  10  in.  in  diameter,  at  $10.80 
a  square  foot  ? 

4.  What  must  be  the  height  of  a  pole  which,  being  broken  30 
feet  from  the  top,  struck  the  ground  18  feet  from  the  bottom? 

5.  Jan.  1.     I  owe  J.  Bush  $325  due  in  4  months ;  $362.50  due 
in  8  months ;  and  $250  due  in  12  months ;  at  what  date  should  I 
give  my  note  to  settle  the  account  ? 

6.  What  is  the  equated  time  of  payment  of  the  following  bill : 
1867.  Walter  Hickok  to  H.  Walcot. 

June  1.     Mdse, $225 

"  12.        "      (4mos.) 250 

Aug.  16.     Cash, 125 

7.  What  is  the  equated  time  of  settling  the  following  account : 
Dr.  A.  Knapp.  Or. 


1867. 

June  1.  Mdse,     .     .     .  $200 

"   16.  "     (3  mos.)      400 

Oct.  20.  Cash,     ...     175 


1867. 

July     4.  Cash,  .     .     .  $200 

Aug.  20.  Mdse,  .    ...     76 

Sept.  20.  "  ...     250 


8.  J.  Smith  has  a  horse  worth  $250,  but  in  trading,  values  it 
at  $280 ;  W.  Read's  horse  cost  $300£ ;  at  what  should  he  value 
it  in  trading  with  Smith  ? 

9.  If  24  men  can  build  a  wall  33f  feet  long,  5f  ft.  high,  and  3£ 
ft.  thick,  in  126  days, by  working  9h.  20min.  each  day;  how  many 
hours  a  day  must  217  men  work  to  build  a  wall  23^  ft.  long,  3f  ft. 
high,  and  2£  ft.  thick,  in  3§  days  ? 

10.  Bought  4  tubs  of  lard,  each  weighing  50  Ibs.,  at  13  cts.  a 
pound;  10  tubs  of  40  Ibs.  each,  at  10  cts. ;  24  tubs,  25  Ibs,  each, 


MISCELLANEOUS    RULES.  285 

at  7  cts. ;  sold  the  whole  at  an  average  of  9^  cts.  a  pound ;  how 
much  was  the  gain? 

11.  A  butcher  bought  lambs  worth  $2,  $2-£,  $3,  and  $4  a  head, 
for  which  he  gave,  on  an  average,  $2^ ;  how  many  at  each  price 
did  he  buy  ? 

12.  A  butcher  bought  12  calves  at  $6  a  head ;  how  many  must 
he  buy  at  $9  and  $15  a  head  that  he  may  sell  them  all  at  $12  a 
head  without  loss  ? 

13.  A  butcher  bought  85  sheep  at  an  average  price  of  $lf  a 
head ;  for  some  he  paid  $1£,  for  some  $1£,  for  some  $2£,  and  for 
others  2^ ;  how  many  at  each  price  did  he  buy  ? 

14.  The  diameter  of  a  circle  is  10  feet ;  what  will  be  the  diame- 
ter of  another  circle  twice  the  area  of  the  first  ? 

15.  A  farmer  wishes  to  make  a  bin  which  will  contain  250 
bushels  of  grain  ;  its  width  to  be  twice  its  depth,  and  its  length 
twice  its  width ;  what  must  be  its  dimensions  ? 

16.  A  boy  agreed  to  work  19  days,  for  which  he  was  to  receive 
4  cents  the  first  day,  and  3  cents  more  every  day  than  the  preced- 
ing ;  how  much  did  he  receive  the  last  day  ? 

17.  A  boy  bought  ten  apples  ;  for  the  first  he  agreed  to  pay  1 
mill,  for  the  second  2  mills,  and  so  on  ;  what  did  he  pay  for  the 
last? 

18.  The  circumference  of  a  park  is  84  rods;  what  is  its  area? 

19.  A  cistern  is  6  feet  deep  and  5^  feet  in  diameter ;  how  many 
hogsheads  of  water  will  it  contain? 

20.  "What  is  the  solidity  of  the  largest  ball  that  can  be  cut  out 
of  a  cubical  block  whose  sides  are  five  inches  square? 

21.  Two   boys   are   running  around  a  block — the  larger  boy 
runs  around  it  every  5-£  minutes,  and  the  smaller  boy  every  6£ 
minutes ;  if  they  started  together  how  many  times  must  each 
run  around  the  block  before  they  will  be  together. 

22.  How  much  corn  must  I  take  to  a  mill,  that  there  may  be 
4  bushels  left  after  taking  4  %  from  each  bushel  for  toll? 

23.  A  liquor  dealer  has  60  gallons  of  brandy,  worth  $3  a  gal- 
-on,  which  he  wishes  to  reduce  so  that  he  can  sell  it  at  $2.40  a 
gallon  ;  how  much  water  must  he  add  to  it  ? 


286  MISCELLANEOUS    RULES. 

24.  A  thief  started  from  a  place  at  midnight  and  traveled  8 
miles  an  hour ;  the  sheriff  started  in  pursuit  3  hours  later,  and 
traveled  10  miles  an  hour;   at  what  time  was  the  thief  over- 
taken ? 

25.  J.  Taylor  can  mow  4  acres  in  3  days,  and  his  son  can  mow 
5  acres  in  4  days ;  in  how  many  days  can  they  both  mow  12|^ 
acres  ? 

26.  I  mixed  16  Ibs.  of  tea  at  75  cents,  20  Ibs.  at  87£  cents,  and 
12  Ibs.  at  $1.25 ;  what  is  a  pound  of  the  mixture  worth? 

27.  If  1£  Ibs.  of  tea  be  worth  8|  Ibs.  of  coffee,  3£  Ibs.  of  coffee 
be  worth  5§  Ibs.  of  sugar,  and  3  Ibs.  of  sugar  be  worth  40  cents, 
what  is  the  price  of  the  tea  ? 

28.  What  is  the  equated  time  of  settling  the  following  account : 

W.  JOHNSON. 

Dr.  Or. 

1868.  1868. 


Jan.  1,  .  Mdse $224 

Feb.  1,  .  Draft  (3  days)     .     182 
"  20,  .  Cash  .     .      .     .      116 


Jan.  20,  .  Cash     ....  $280 

Feb.  6,  .  Draft  (10  days).      132 

"    25,  .  Consm't  ...     450 


29.  A  grocer  has  spices  at  9d.,  Is.,  2s.,  2s.  6d. ;  how  must  he 
mix  them  so  that  he  can  sell  the  mixture  at  Is.  8d.  a  pound  ? 

30.  What  is  the  greatest  number  of  hills  of  corn  that  can  be 
planted  on  a  square  acre,  the  centers  of  the  hills  to  be  3^  feet 
apart  ? 

31.  If  10   barrels  of  water  flow  through  a  pipe  2-|  inches  in 
diameter,  what  must  be  the  diameter  of  a  pipe  that  will  discharge 
four  times  as  much  in  the  same  time  ? 

32.  If  a  silver  ball  3  inches  in  diameter  be  worth  $270,  what 
is  another  6  inches  in  diameter  worth? 

33.  A  man  bought  19  yards  of  linen ;  for  the  first  he  gave  Is., 
and  for  the  last  £1  17s. ;  what  did  the  whole  cost? 

34.  What  is  the  area  of  a  circular  plot,  11  rods  in  diameter? 

35.  What  are  the  solid  contents  of  a  column  whose  average 
diameter  is  18  inches,  and  whose  length  is  20  feet? 


QUESTIONS. 


Article  1.— What  is  arithmetic?  Number?  Abstract  numbers? 
Concrete  ? 

[2]  What  is  notation  ?  How  many  kinds  of  notation  are  in  com- 
mon use  ?  [3]  What  is  the  Roman  method?  How  many  letters  does 
it  use  ?  What  is  the  effect  of  repeating  the  letters  ?  of  writing  a  letter 
before  another  of  greater  value  ?  after  it  ?  [Examples.  ] 

[4]  What  is  the  Arabic  notation  ?  How  many  and  what  figures 
does  it  employ  ?  Which  are  called  digits  ?  What  is  the  simple  value 
of  a  figure?  [Examples.]  What  is  its  value  when  written  before  anoth- 
er? [Examples.]  When  written  before  two  others?  three,  four,  etc.  ? 
How  do  figures  increase  in  value  ?  Kepeat  the  French  Notation  and 
Numeration  table.  How  is  it  divided  ?  Name  the  periods  in  order. 
How  may  numbers  consisting  of  several  figures  be  written,  or  what  is 
a  rule  for  Notation? 

[Instead  of  learning  the  rules  in  the  book,  it  is  better  that  the  pupil  should 
thoroughly  understand  them,  and  express  the  ideas  in  his  own  words.] 

[5]  What  is  numeration  ?  How  may  large  numbers  be  read  ? 

[6]  What  are  the  fundamental  rules  of  arithmetic  ? 

[7]  What  is  addition?  What  is  the  number  found,  or  answer 
called?  What  is  simple  addition?  [Illustration.]  What  is  the  sign 
of  addition  ?  Give  a  rule  for  adding  numbers,  consisting  of  one  fig- 
ure or  units  ;  of  units,  tens,  etc. ,  or  several  figures. 

[8]  What  is  subtraction?  What  is  the  answer  called?  the  num- 
ber to  be  subtracted  ?  the  number  from  which  another  is  subtracted? 
What  is  simple  subtraction?  [Illustration.]  What  is  the  sign  of 
subtraction  ?  [Illustration.  ]  Give  a  rule  for  subtraction. 

[9]  What  is  multiplication?  What  is  the  number  to  be  multi- 
plied called  ?  the  number  by  which  another  is  multiplied  ?  the  an- 
swer ?  What  are  the  factors  ?  What  is  simple  multiplication  ?  the 
sign  ?  What  two  numbers  in  multiplication  are  properly  of  the  same 
name  ?  What  kind  of  a  number  is  the  multiplier  ?  [Illustration.  ] 

[10]  Give  a  rule  when  the  multiplier  consists  of  one  figure  or  a 
numberless  than  12  :  [12]  when  it  is  greater  than  12  :  [13]  for  the 


288  QUESTIONS. 

multiplication  of  numbers  having  ciphers  on  the  right ;  [14]  What 
is  a  composite  number  ?  [Illustration.  ]  Give  a  rule  for  multiplying 
by  composite  numbers. 

[15]  What  is  division?  What  is  the  number  to  be  divided 
called?  the  dividing  number?  the  answer?  the  number  that  is  some- 
times left  ?  •  What  is  the  sign  of  division  ?  In  what  other  way  is 
division  sometimes  expressed  ?  What  is  such  an  expression  called  ? 
With  what  do  the  divisor  and  quotient  correspond  ?  the  dividend  ? 
With  what  must  the  name  of  one  of  the  factors  in  division  corres- 
pond? What  is  the  other  factor? 

[16]  How  many  kinds  of  division  are  there  ?  What  is  short  divi- 
sion? long  division?  Give  a  rule  for  short  division;  [17]  long 
division.  [18]  When  there  are  ciphers  on  the  right  hand  of  the 
divisor  ;  [19]  when  the  divisor  is  a  composite  number. 

[20]  How  is  the  quotient  affected  when  the  dividend  is  multi- 
plied by  any  number  ?  when  the  divisor  is  divided  ?  when  the  divi- 
dend is  divided?  when  the  divisor  is  multiplied?  when  the  divi- 
dend and  divisor  are  both  multiplied  or  both  divided  by  the  same 
number  ? 

United  States  Money.  [22]  What  is  United  States  Money  ?  Of 
what  does  it  consist?  What  are  its  coins  in  gold?  silver?  Repeat 
the  table.  [23]  What  are  aliquot  parts?  What  part  of  $1  are  10 
cents?  12|?  16$?  20?  25?  33£?  37i?  50?  62£?  75?  87i? 
[24]  Give  a  rule  for  writing  United  States  Money.  [25]  for  read- 
ing United  States  Money.  [26]  for  reducing  dollars  to  cents; 
cents  to  mills  ;  dollars  to  mills  ;  dollars  and  cents  to  cents  ;  mills  to 
cents  ;  cents  to  dollars  ;  mills  to  dollars. 

[27]  In  what  respect  is  United  States  Money  like  simple  num- 
bers? [28]  Rule  for  addition  of  United  States  Money  ;  [29]  sub- 
traction ;  [30]  multiplication ;  [31]  division ;  [32]  when  the 
price  of  anything  is  an  aliquot  part  of  $1.  [34)  when  the  price  is 
per  hundred  or  thousand  ? 

Compound  Numbers.  [36]  What  are  compound  numbers?  [Illus- 
tration. ]  What  are  the  general  names  ?  [37]  What  is  English  or 
Sterling  Money?  Eepeat  the  table.  [38]  For  what  is  Troy  Weight 
used?  Table.  [39]  Avoirdupois  Weight?  Table.  [40]  Apotheca- 
ries Weight?  Table.  [41]  How  many  pounds  in  a  barrel  of  flour? 
of  beef,  pork,  or  fish?  firkin  of  butter?  bushel  of  wheat?  rye  or 
corn?  barley?  oats?  [42]  What  is  Cloth  Measure?  Table. 
[43]  Long  Measure?  Table.  [44]  Surveyor's  Measure?  Table. 


QUESTIONS.  289 

[45]  Square  Measure  ?  Table.  What  is  a  square  ?  How  may  the 
contents  of  a  square  be  found  ?  What  is  the  difference  between  3 
square  feet  and  3  feet  square,  etc.,  etc.?  What  is  a  rectangle? 
How  may  the  contents  of  a  rectangle  be  found?  [46]  Cubic  Mea- 
sure? Table.  What  is  a  cube?  How  may  the  contents  of  a  cube 
be  found?  How  may  one  of  the  dimensions  be  found?  [47]  Wine 
Measure?  Table.  [48]  Beer  Measure?  Table.  [49]  Dry  Measure? 
Table.  [50]  Time  Measure?  Table.  [51]  Circular  Measure?  Ta- 
ble. What  is  a  circle?  the  circumference?  diameter?  radius? 
[52]  Miscellaneous  Table  of  units,  etc. ,  paper  and  books  ? 

Reduction  of  Compound  Numbers.  [53]  What  is  reduction  of 
compound  numbers  ?  What  two  kinds  are  there  ?  [54]  What  is  re- 
duction descending  ?  ascending  ?  Give  a  rule  for  reduction  descend- 
ing; ascending. 

[68]  Give  a  rule  for  addition  of  Compound  Numbers.  [69]  Sub- 
traction. [71]  Multiplication.  [72]  Division. 

[74]  Give  a  rule  for  finding  the  difference  of  time  when  the  dif- 
ference of  longitude  is  known,  [75]  for  finding  the  difference  of 
longitude  when  the  difference  of  time  is  known. 

Cancellation.  [77]  What  is  cancellation?  Why  may  we  cancel? 
Give  a  rule  for  cancellation. 

Properties  of  Numbers.  [78]  What  are  even  numbers?  odd? 
prime  ?  composite  ?  [79]  Give  a  rule  for  resolving  composite  num- 
bers into  prime  factors. 

[81]  What  is  the  greatest  common  divisor  of  two  or  more  num- 
bers? [82]  How  may  it  be  found?  [83]  What  is  the  multiple  of  a 
number  ?  the  common  multiple  of  two  or  more  numbers  ?  the  least 
common  multiple  ?  How  may  it  be  found  ?  Rule  I.,  II.,  III.,  IV. 

Fractions.  [86]  What  are  fractions?  What  is  meant  by  £,  £,  £? 
[87]  How  many  kinds  of  fractions  are  there?  What  are  common 
fractions?  decimal  fractions?  [88]  How  are  common  fractions 
written  ?  What  is  the  number  above  the  line  called  ?  the  number 
below  the  line  ?  What  are  the  terms  of  a  fraction  ?  What  does  the 
denominator  show?  With  what  does  it  correspond  in  division? 
What  does  the  numerator  show  ?  With  what  does  it  correspond  in 
division  ? 

[89]  How  are  common  fractions  divided  ?    What  is  a  simple  frac- 
tion? when  is  it  proper?  when  improper?    What  is  a  compound 
fraction  ?  complex  ?  mixed  number  ?    What  are  like  fractions  ?  un- 
like !  [90]  What  is  the  value  of  a  fraction?  [91]  How  is  the  value 
12 


290  QUESTIONS.  . 

of  a  fraction  affected  by  multiplying  the  numerator  by  any  number  ? 
dividing  the  denominator?  dividing  the  numerator?  multiplying 
the  denominator?  multiplying  both  the  numerator  and  denominator? 
dividing  both  ? 

[92]  What  is  reduction  of  fractions?  What  are  their  simplest 
forms?  [93]  When  is  a  fraction  in  its  lowest  terms?  How  may  it 
be  reduced  to  its  lowest  tertns  ?  [94]  To  what  may  an  improper  frac- 
tion be  reduced  ?  How  ?  [95]  To  what  may  a  mixed  number  be  re- 
duced ?  How  ?  How  may  a  whole  number  be  reduced  to  a  fraction  ? 
[96,  97]  How  may  unlike  fractions  be  reduced  to  like  fractions  ? 
[Rules  L,  II.]  [98]  In  what  is  reduction  of  compound  fractions  in- 
cluded. 

[100]  Give  a  rule  for  addition  of  fractions?  [101]  subtraction? 
[102,  103,  104]  multiplication?  [106,  107]  division?  [108]  In 
what  is  reduction  of  complex  fractions  included  ?  In  what  respects  ? 

Decimal  Fractions.  [109]  What  are  decimal  fractions  ?  How  are 
they  distinguished  from  whole  numbers?  [110]  Repeat  the  table. 
[Ill]  What  is  the  denominator  of  a  decimal  fraction  ?  What  is  the 
effect  of  prefixing  a  cipher?  why?  annexing  a  cipher?  why? 
Bead  the  examples.  [112]  Eule  for  writing  decimals,  examples. 
[113]  Eule  for  addition  of  decimals  ?  [114]  subtraction  ?  [115]  What 
is  the  general  principle  of  multiplication  of  decimals  ?  Eule  ?  Eule 
for  multiplying  by  10,  100  etc.  ?  [116]  What  is  the  general  principle 
of  division  of  decimals  ?  Eule  ?  Eule  for  dividing  by  10,  100  etc.  ? 
[118]  How  may  common  fractions  be  reduced  to  decimals?  [119] 
decimal  to  common  fractions  ? 

[120]  Fractional  Compound  lumbers,  what  arc  they  ?  Examples. 
General  Eule.  [121]  How  may  compound  whole  numbers  be  re- 
duced to  fractions  ?  [122]  How  may  fractional  compound  numbers 
be  reduced  to  whole  numbers  ?  [123]  to  other  denominations  ? 

[127]  Duodecimals,  what  are  they?  Whence  arise?  From  what 
is 'the  name  derived?  How  added,  subtracted  etc.  ?  What  denomi- 
nation is  the  product  of  feet  by  feet  ?  square  feet  by  feet  ?  feet  by 
inches  ?  square  feet  by  inches  ?  inches  by  inches  ?  square  inches  by 
inches?  inches  by  seconds?  square  inches  by  seconds?  seconds  by 
seconds  ? 

[128]  What  is  Analysis  ?    General  process  ? 

[129]  What  is  Percentage  ?  From  what  is  the  term  per  cent,  de- 
rived ?  [130]  What  three  things  are  chiefly  considered  in  percentage  ? 
IThatis  the  principal?  rate?  percentage?  How  is  the  rate  expressed ? 


QUESTIONS.  291 

examples.  [131]  How  may  the  percentage  be  found  ?  [132]  the  rate 
per  cent.  ?  [133,  134]  the  principal  ? 

[136]  Applications  of  Percentage.  To  what  is  percentage  appli- 
cable ?  [137]  What  is  Commission  ?  A.  consignment  ?  gross  pro- 
ceeds ?  net  proceeds  ?  [138]  an  account  of  sales  ?  [139]  brokerage  ?  a 
broker  ?  [110]  stocks  ?  common  value  of  a  share  ?  when  are  stocks 
at  par  ?  above  par,  at  a  premium  or  an  advance?  below  par  or  at  a 
discount?  stockholders?  dividend?  bonds? 

[141]  How  is  gold  bought  and  sold  ?  [142]  What  is  insurance  ?  fire 
insurance?  marine  insurance?  life  insurance?  a  policy?  the  pre- 
mium? [144]  What  is  profit  and  loss  ?  how  estimated? 

[145]  Interest,  what  is  it  ?  What  is  the  principal  when  interest  is 
to  be  found?  rate?  amount?  simple  interest?  legal  interest?  the 
legal  interest  in  the  different  States  ?  How  is  interest  found  for  one 
year?  two  or  more  years ?  months?  days?  [146]  Another  method ? 
[147]  Exact  interest?  [148]  How  is  interest  on  Sterling  Money 
calculated  ? 

[149]  Partial  Payments,  what  are  they?  What  is  a  note  ?  Write  a 
note  in  the  usual  form.  Who  is  the  maker  or  drawer?  the  payee? 
What  is  the  face  of  a  note  ?  How  is  the  amount  due  on  a  note  found 
when  partial  payments  have  been  made  ?  [150]  Merchants'  Rule  ? 
[151]  Connecticut  rule  ? 

[152]  How  is  the  rate  found  from  the  principal,  interest  and  time  ? 
[153]  The  time  from  the  principal,  interest  and  rate?  [154]  The 
principal  from  the  interest,  rate  and  time  ? 

[155]  Compound  interest,  what  is  it  ?  how  found? 

[156]  Discount,  what  is  it  ?  What  is  the  present  worth  of  a  sum 
or  debt?  How  is  the  present  worth  found?  How  is  the  discount 
found? 

[157]  Bank  discount,  what  is  it  ?  How  does  it  differ  from  true 
discount  ?  How  may  it  be  found  ?  [158]  How  may  the  amount  of  a 
note  be  found,  which  will  be  worth  a  given  sum  at  bank  discount? 

[159]  Taxes.  What  is  a  tax?  poll  tax?  income  tax?  What  is  real 
estate  ?  personal  property  ?  an  inventory  or  list  ?  explain  by  an  ex- 
ample the  process  of  finding  a  person's  tax. 

[160]  Duties,  what  are  they?  What  is  a  port  of  entry?  a  custom 
house?  ad  valorem  duty  ?  specific  duty?  an  in  voice?  tare?  draft? 
leakage  ?  gross  weight  ?  net  weight  ?  how  is  the  duty  on  goods  found  ? 

[161]  Exchange,  what  is  it?  domestic?  foreign?  write  a  draft. 
Who  is  the  drawer  ?  drawee  ?  payee  ?  How  may  a  draft  be  accepted  ? 


292  QUESTIONS. 

What  is  an  acceptance?  How  may  a  draft  be  indorsed?  What  is  the 
use  of  an  indorsement? 

[162]  Why  is  the  exchange  on  England  always  at  a  premium  ?  [165] 
how  may  it  be  found  ?  how  may  we  find  the  amount  of  a  bill  of  ex- 
change which  can  be  bought  with  U.  S .  money  ? 

[165]  Partnership,  what  is  it  ?  a  firm  or  house  ?  [166]  How  may 
each  partner's  share  of  profit  and  loss  be  found?  [167]  when  the 
times  are  unequal? 

[169]  Equation  of  Payments  is  what?  equated  time?  an  account 
current?  [170]  How  may  the  equated  time  be  found?  [171]  when 
partial  payments  have  been  made?  [172]  of  an  account  current? 
[173]  of  an  account  bearing  interest  ? 

[174]  Reduction  of  Currencies,  what  is  it  ?  Why  have  State  cur- 
rencies been  used?  Wha't  is  New  England  Currency?  its  value? 
New  York  Currency?  its  value?  Pennsylvania  Currency?  its  value? 
Georgia  Currency?  its  value?  [175]  How  may  U.  S.  money  be  re- 
duced to  State  currencies?  [176]  How  may  State  currencies  be  re- 
duced to  U.  S.  money? 

[178]  Ratio,  what  is  it?  arithmetical?  geometrical?  To  what  is 
the  ratio  of  two  numbers  equal?  how  expressed?  What  are  the 
terms  of  a  ratio  ?  What  are  they  both  called  ?  What  is  the  first  term 
caUed?  the  last? 

[179]  Compound  ratio,  what  is  it? 

[180]  Proportion,  what  is  it  ?  how  expressed?  what  are  the  ex- 
tremes ?  the  means  ?  what  is  a  mean  proportional?  [181]  a  direct  pro- 
portion? an  indirect  or  inverse  proportion?  How  is  the  product  of 
the  extremes  compared  with  that  of  the  means?  How  may  the  terms 
be  arranged  ?  How  may  the  required  term  be  found  ? 

[183]  Compound  Proportion,"what  is  it?  How  may  the  terms  be 
arranged?  How  may  the  required  term  be  found? 

[184]  Conjoined  Proportion,  what  is  it  ?  How  may  the  terms  be 
arranged  ?  How  may  the  required  term  be  found  ? 

[186]  Alligation,  what  is  it?  [187]  What  is  alligation  medial? 
How  is  the  price  of  a  mixture  found  ?  [188]  What  is  alligation  alter- 
nate ?  How  may  the  quantities  of  different  ingredients  be  found, 
that  will  form  a  mixture  at  a  certain  price  ? 

[191]  Involution,  what  is  it?  What  is  a  power?  How  are  differ- 
ent powers  distinguished  ?  What  is  the  second  power  usually  called  ? 
the  third  ?  How  are  powers  found  ? 


QUESTIONS.  293 

[192]  Evolution)  what  is  it  ?  What  is  a  root  ?  a  square  root '(  cube 
root  ?  the  radical  sign  ?  How  are  the  different  roots  expressed  ? 

[193]  What  is  the  square  root  of  a  number?  of  1?  4?  9?&c.? 
How  may  we  find  the  square  root  of  any  number  ? 

[194]  How  is  square  root  applicable  to  circles,  squares,  or  other 
similar  figures  ?  What  is  a  triangle  ?  a  right  angle  ?  a  right  angled 
triangle?  the  hypothenuse?  base?  perpendicular?  How  may  the 
hypothenuse  be  found  ?  the  base  or  perpendicular  ?  [195]  the  side  of 
a  square  ?  [196]  the  areas  of  similar  figures  ? 

[197]  What  is  the  cube  root  of  a  number  ?  of  1?  8  ?  27  ?  64  ?  125,  &c.  ? 
How  may  the  cube  root  of  any  number  be  found  ?  [198]  How  is 
cube  root  applicable  to  solids  ? 

[199]  Progression,  what  is  it?  what  kinds  are  there?  What  are 
the  terms  ?  the  extremes  ?  the  means  ?  an  ascending  series  ?  descend- 
ing? 

[200]  Arithmetical  Progression,  what  is  it?  What  are  to  bo 
found?  [201]  How  may  the  last  term  be  found?  [202]  the  common 
difference?  [203]  the  number  of  terms?  [204]  the  sum  of  the, 


[205]  Geometrical  Progression,  what  is  it?  What  are  to  be 
found?  [206]  How  may  the  last  term  be  found?  [207]  the  sum  of 
the  series? 

[208]  Mensuration,  what  is  it?  What  is  a  parallelogram?  cylin- 
der ?  prism  ?  pyramid  ?  cone  ?  frustum  ?  sphere  ?  How  is  the  sur- 
face of  each  found  ?  How  are  the  solid  contents  found  ? 


. 


UNIVERSITY    OF    CALIFORNIA 
LIBRARY 

Due  two  weeks  after  date. 


30m- 7,' 12 


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